topic_name stringclasses 721
values | problem stringlengths 4 1.36k ⌀ | hints/solutions stringlengths 5 3.3k |
|---|---|---|
307 | Solve for $a$, $ \dfrac{10}{3a} = -\dfrac{2a - 3}{15a} - \dfrac{10}{3a} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3a$ $15a$ and $3a$ The common denominator is $15a$ To get $15a$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{10}{3a} \times \dfrac{5}{5} = \dfrac{50}{15a} $ The denominator... |
307 | Solve for $y$, $ \dfrac{5}{3y} = -\dfrac{4}{9y} - \dfrac{y - 1}{3y} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3y$ $9y$ and $3y$ The common denominator is $9y$ To get $9y$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{5}{3y} \times \dfrac{3}{3} = \dfrac{15}{9y} $ The denominator of t... |
307 | Solve for $r$, $ \dfrac{7}{25r} = -\dfrac{8}{10r} + \dfrac{5r + 2}{5r} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25r$ $10r$ and $5r$ The common denominator is $50r$ To get $50r$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{7}{25r} \times \dfrac{2}{2} = \dfrac{14}{50r} $ To get $50r$ i... |
307 | Solve for $q$, $ \dfrac{4q + 4}{4q - 1} = \dfrac{7}{20q - 5} - \dfrac{4}{20q - 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4q - 1$ $20q - 5$ and $20q - 5$ The common denominator is $20q - 5$ To get $20q - 5$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{4q + 4}{4q - 1} \times \dfrac{5}{5} = \dfr... |
307 | Solve for $z$, $ -\dfrac{1}{5z + 5} = -\dfrac{z - 7}{5z + 5} + \dfrac{5}{5z + 5} $ | If we multiply both sides of the equation by $5z + 5$ , we get: $ -1 = -z + 7 + 5$ $ -1 = -z + 12$ $ -13 = -z $ $ z = 13$ |
307 | Solve for $a$, $ -\dfrac{3}{a} = \dfrac{5}{4a} + \dfrac{4a - 5}{a} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $a$ $4a$ and $a$ The common denominator is $4a$ To get $4a$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{3}{a} \times \dfrac{4}{4} = -\dfrac{12}{4a} $ The denominator of th... |
307 | Solve for $x$, $ -\dfrac{2x - 4}{2x - 4} = \dfrac{8}{2x - 4} - \dfrac{2}{8x - 16} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2x - 4$ $2x - 4$ and $8x - 16$ The common denominator is $8x - 16$ To get $8x - 16$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{2x - 4}{2x - 4} \times \dfrac{4}{4} = -\df... |
307 | Solve for $x$, $ -\dfrac{4x}{2x} = \dfrac{4}{10x} + \dfrac{10}{4x} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2x$ $10x$ and $4x$ The common denominator is $20x$ To get $20x$ in the denominator of the first term, multiply it by $\frac{10}{10}$ $ -\dfrac{4x}{2x} \times \dfrac{10}{10} = -\dfrac{40x}{20x} $ To get $... |
307 | Solve for $q$, $ -\dfrac{1}{20q^2} = -\dfrac{4q + 6}{5q^2} + \dfrac{7}{15q^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $20q^2$ $5q^2$ and $15q^2$ The common denominator is $60q^2$ To get $60q^2$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ -\dfrac{1}{20q^2} \times \dfrac{3}{3} = -\dfrac{3}{60q^2} $... |
307 | Solve for $a$, $ \dfrac{2}{a^2} = \dfrac{9}{3a^2} + \dfrac{3a + 6}{4a^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $a^2$ $3a^2$ and $4a^2$ The common denominator is $12a^2$ To get $12a^2$ in the denominator of the first term, multiply it by $\frac{12}{12}$ $ \dfrac{2}{a^2} \times \dfrac{12}{12} = \dfrac{24}{12a^2} $ T... |
307 | Solve for $p$, $ \dfrac{2}{4p - 5} = -\dfrac{5p + 3}{4p - 5} + \dfrac{9}{4p - 5} $ | If we multiply both sides of the equation by $4p - 5$ , we get: $ 2 = -5p - 3 + 9$ $ 2 = -5p + 6$ $ -4 = -5p $ $ p = \dfrac{4}{5}$ |
307 | Solve for $a$, $ -\dfrac{3a + 10}{2a + 2} = -\dfrac{2}{2a + 2} + \dfrac{1}{10a + 10} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2a + 2$ $2a + 2$ and $10a + 10$ The common denominator is $10a + 10$ To get $10a + 10$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{3a + 10}{2a + 2} \times \dfrac{5}{5} = ... |
307 | Solve for $x$, $ -\dfrac{9}{x + 3} = -\dfrac{5}{x + 3} - \dfrac{x + 9}{x + 3} $ | If we multiply both sides of the equation by $x + 3$ , we get: $ -9 = -5 - x - 9$ $ -9 = -x - 14$ $ 5 = -x $ $ x = -5$ |
307 | Solve for $q$, $ \dfrac{5}{2q^2} = -\dfrac{1}{q^2} - \dfrac{q - 7}{5q^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2q^2$ $q^2$ and $5q^2$ The common denominator is $10q^2$ To get $10q^2$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{5}{2q^2} \times \dfrac{5}{5} = \dfrac{25}{10q^2} $ To g... |
307 | Solve for $q$, $ -\dfrac{3q - 8}{q - 4} = -\dfrac{7}{3q - 12} - \dfrac{6}{4q - 16} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $q - 4$ $3q - 12$ and $4q - 16$ The common denominator is $12q - 48$ To get $12q - 48$ in the denominator of the first term, multiply it by $\frac{12}{12}$ $ -\dfrac{3q - 8}{q - 4} \times \dfrac{12}{12} =... |
307 | Solve for $x$, $ -\dfrac{8}{x} = -\dfrac{7}{x} + \dfrac{5x + 2}{2x} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $x$ $x$ and $2x$ The common denominator is $2x$ To get $2x$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{8}{x} \times \dfrac{2}{2} = -\dfrac{16}{2x} $ To get $2x$ in the de... |
307 | Solve for $p$, $ \dfrac{6}{5p + 15} = \dfrac{9}{p + 3} + \dfrac{p - 4}{3p + 9} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5p + 15$ $p + 3$ and $3p + 9$ The common denominator is $15p + 45$ To get $15p + 45$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{6}{5p + 15} \times \dfrac{3}{3} = \dfrac{1... |
307 | Solve for $z$, $ -\dfrac{7}{15z + 12} = -\dfrac{3z - 6}{5z + 4} + \dfrac{10}{5z + 4} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15z + 12$ $5z + 4$ and $5z + 4$ The common denominator is $15z + 12$ The denominator of the first term is already $15z + 12$ , so we don't need to change it. To get $15z + 12$ in the denominator of the s... |
307 | Solve for $t$, $ -\dfrac{5}{t - 5} = -\dfrac{3}{t - 5} - \dfrac{3t}{t - 5} $ | If we multiply both sides of the equation by $t - 5$ , we get: $ -5 = -3 - 3t$ $ -5 = -3t - 3$ $ -2 = -3t $ $ t = \dfrac{2}{3}$ |
307 | Solve for $r$, $ -\dfrac{4r - 7}{r + 3} = \dfrac{4}{r + 3} - \dfrac{1}{r + 3} $ | If we multiply both sides of the equation by $r + 3$ , we get: $ -4r + 7 = 4 - 1$ $ -4r + 7 = 3$ $ -4r = -4 $ $ r = 1$ |
307 | Solve for $y$, $ -\dfrac{3}{25y + 10} = -\dfrac{10}{10y + 4} + \dfrac{4y - 1}{5y + 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25y + 10$ $10y + 4$ and $5y + 2$ The common denominator is $50y + 20$ To get $50y + 20$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{3}{25y + 10} \times \dfrac{2}{2} = -\d... |
307 | Solve for $a$, $ \dfrac{10}{a^2} = -\dfrac{6}{a^2} + \dfrac{5a - 7}{a^2} $ | If we multiply both sides of the equation by $a^2$ , we get: $ 10 = -6 + 5a - 7$ $ 10 = 5a - 13$ $ 23 = 5a $ $ a = \dfrac{23}{5}$ |
307 | Solve for $t$, $ -\dfrac{t}{t^3} = -\dfrac{2}{3t^3} + \dfrac{7}{t^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $t^3$ $3t^3$ and $t^3$ The common denominator is $3t^3$ To get $3t^3$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ -\dfrac{t}{t^3} \times \dfrac{3}{3} = -\dfrac{3t}{3t^3} $ The den... |
307 | Solve for $k$, $ -\dfrac{k + 1}{25k^3} = -\dfrac{3}{5k^3} - \dfrac{8}{20k^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25k^3$ $5k^3$ and $20k^3$ The common denominator is $100k^3$ To get $100k^3$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{k + 1}{25k^3} \times \dfrac{4}{4} = -\dfrac{4k + ... |
307 | Solve for $t$, $ -\dfrac{10}{2t - 3} = -\dfrac{t - 5}{10t - 15} - \dfrac{3}{2t - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2t - 3$ $10t - 15$ and $2t - 3$ The common denominator is $10t - 15$ To get $10t - 15$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{10}{2t - 3} \times \dfrac{5}{5} = -\dfr... |
307 | Solve for $a$, $ \dfrac{a - 7}{2a} = -\dfrac{2}{2a} - \dfrac{9}{2a} $ | If we multiply both sides of the equation by $2a$ , we get: $ a - 7 = -2 - 9$ $ a - 7 = -11$ $ a = -4 $ |
307 | Solve for $x$, $ -\dfrac{6}{3x - 5} = \dfrac{x + 10}{6x - 10} + \dfrac{1}{3x - 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3x - 5$ $6x - 10$ and $3x - 5$ The common denominator is $6x - 10$ To get $6x - 10$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{6}{3x - 5} \times \dfrac{2}{2} = -\dfrac{1... |
307 | Solve for $k$, $ -\dfrac{2}{12k - 4} = -\dfrac{k - 5}{6k - 2} + \dfrac{9}{3k - 1} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $12k - 4$ $6k - 2$ and $3k - 1$ The common denominator is $12k - 4$ The denominator of the first term is already $12k - 4$ , so we don't need to change it. To get $12k - 4$ in the denominator of the secon... |
307 | Solve for $n$, $ -\dfrac{3}{n} = -\dfrac{n - 3}{5n} - \dfrac{4}{n} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $n$ $5n$ and $n$ The common denominator is $5n$ To get $5n$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{3}{n} \times \dfrac{5}{5} = -\dfrac{15}{5n} $ The denominator of th... |
307 | Solve for $k$, $ -\dfrac{6}{15k} = -\dfrac{5k - 9}{20k} - \dfrac{6}{5k} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15k$ $20k$ and $5k$ The common denominator is $60k$ To get $60k$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{6}{15k} \times \dfrac{4}{4} = -\dfrac{24}{60k} $ To get $60k$... |
307 | Solve for $y$, $ -\dfrac{9}{9y^2} = -\dfrac{5y - 4}{9y^2} - \dfrac{7}{3y^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $9y^2$ $9y^2$ and $3y^2$ The common denominator is $9y^2$ The denominator of the first term is already $9y^2$ , so we don't need to change it. The denominator of the second term is already $9y^2$ , so we ... |
307 | Solve for $t$, $ \dfrac{7}{t + 1} = \dfrac{2}{t + 1} - \dfrac{5t - 6}{t + 1} $ | If we multiply both sides of the equation by $t + 1$ , we get: $ 7 = 2 - 5t + 6$ $ 7 = -5t + 8$ $ -1 = -5t $ $ t = \dfrac{1}{5}$ |
307 | Solve for $n$, $ -\dfrac{5}{4n} = -\dfrac{4}{20n} - \dfrac{n + 6}{4n} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4n$ $20n$ and $4n$ The common denominator is $20n$ To get $20n$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{5}{4n} \times \dfrac{5}{5} = -\dfrac{25}{20n} $ The denominato... |
307 | Solve for $a$, $ -\dfrac{8}{8a^2} = \dfrac{4a - 3}{4a^2} + \dfrac{3}{20a^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8a^2$ $4a^2$ and $20a^2$ The common denominator is $40a^2$ To get $40a^2$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{8}{8a^2} \times \dfrac{5}{5} = -\dfrac{40}{40a^2} $ ... |
307 | Solve for $k$, $ \dfrac{4}{4k} = -\dfrac{2}{4k} + \dfrac{4k - 6}{4k} $ | If we multiply both sides of the equation by $4k$ , we get: $ 4 = -2 + 4k - 6$ $ 4 = 4k - 8$ $ 12 = 4k $ $ k = 3$ |
307 | Solve for $n$, $ -\dfrac{3}{4n - 3} = \dfrac{n + 6}{4n - 3} + \dfrac{6}{20n - 15} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4n - 3$ $4n - 3$ and $20n - 15$ The common denominator is $20n - 15$ To get $20n - 15$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{3}{4n - 3} \times \dfrac{5}{5} = -\dfra... |
307 | Solve for $y$, $ \dfrac{3}{16y + 8} = -\dfrac{y + 8}{4y + 2} + \dfrac{5}{16y + 8} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $16y + 8$ $4y + 2$ and $16y + 8$ The common denominator is $16y + 8$ The denominator of the first term is already $16y + 8$ , so we don't need to change it. To get $16y + 8$ in the denominator of the seco... |
307 | Solve for $z$, $ -\dfrac{9}{3z - 9} = -\dfrac{8}{z - 3} - \dfrac{5z + 3}{z - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3z - 9$ $z - 3$ and $z - 3$ The common denominator is $3z - 9$ The denominator of the first term is already $3z - 9$ , so we don't need to change it. To get $3z - 9$ in the denominator of the second term... |
307 | Solve for $y$, $ \dfrac{5}{12y} = \dfrac{8}{3y} - \dfrac{5y - 6}{9y} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $12y$ $3y$ and $9y$ The common denominator is $36y$ To get $36y$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{5}{12y} \times \dfrac{3}{3} = \dfrac{15}{36y} $ To get $36y$ in... |
307 | Solve for $p$, $ -\dfrac{1}{p + 2} = -\dfrac{7}{4p + 8} + \dfrac{5p - 6}{4p + 8} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $p + 2$ $4p + 8$ and $4p + 8$ The common denominator is $4p + 8$ To get $4p + 8$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{1}{p + 2} \times \dfrac{4}{4} = -\dfrac{4}{4p ... |
307 | Solve for $p$, $ \dfrac{3p - 2}{8p - 4} = \dfrac{1}{12p - 6} + \dfrac{1}{4p - 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8p - 4$ $12p - 6$ and $4p - 2$ The common denominator is $24p - 12$ To get $24p - 12$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{3p - 2}{8p - 4} \times \dfrac{3}{3} = \df... |
307 | Solve for $x$, $ \dfrac{x + 10}{4x + 16} = \dfrac{7}{x + 4} + \dfrac{1}{x + 4} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4x + 16$ $x + 4$ and $x + 4$ The common denominator is $4x + 16$ The denominator of the first term is already $4x + 16$ , so we don't need to change it. To get $4x + 16$ in the denominator of the second ... |
307 | Solve for $t$, $ \dfrac{1}{5t - 20} = \dfrac{2t - 2}{t - 4} + \dfrac{3}{3t - 12} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5t - 20$ $t - 4$ and $3t - 12$ The common denominator is $15t - 60$ To get $15t - 60$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{1}{5t - 20} \times \dfrac{3}{3} = \dfrac{... |
307 | Solve for $k$, $ \dfrac{9}{8k} = \dfrac{4}{20k} - \dfrac{5k + 3}{4k} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8k$ $20k$ and $4k$ The common denominator is $40k$ To get $40k$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{9}{8k} \times \dfrac{5}{5} = \dfrac{45}{40k} $ To get $40k$ in ... |
307 | Solve for $r$, $ \dfrac{r - 8}{r} = \dfrac{3}{4r} + \dfrac{3}{r} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $r$ $4r$ and $r$ The common denominator is $4r$ To get $4r$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{r - 8}{r} \times \dfrac{4}{4} = \dfrac{4r - 32}{4r} $ The denominato... |
307 | Solve for $r$, $ -\dfrac{1}{12r - 8} = \dfrac{r + 10}{15r - 10} + \dfrac{5}{3r - 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $12r - 8$ $15r - 10$ and $3r - 2$ The common denominator is $60r - 40$ To get $60r - 40$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{1}{12r - 8} \times \dfrac{5}{5} = -\df... |
307 | Solve for $q$, $ \dfrac{q + 1}{3q} = \dfrac{3}{5q} - \dfrac{1}{q} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3q$ $5q$ and $q$ The common denominator is $15q$ To get $15q$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{q + 1}{3q} \times \dfrac{5}{5} = \dfrac{5q + 5}{15q} $ To get $15... |
307 | Solve for $x$, $ \dfrac{3x + 4}{4x - 2} = \dfrac{4}{4x - 2} - \dfrac{10}{8x - 4} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4x - 2$ $4x - 2$ and $8x - 4$ The common denominator is $8x - 4$ To get $8x - 4$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{3x + 4}{4x - 2} \times \dfrac{2}{2} = \dfrac{6... |
307 | Solve for $r$, $ \dfrac{r + 5}{8r^3} = -\dfrac{10}{10r^3} - \dfrac{6}{4r^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8r^3$ $10r^3$ and $4r^3$ The common denominator is $40r^3$ To get $40r^3$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{r + 5}{8r^3} \times \dfrac{5}{5} = \dfrac{5r + 25}{40... |
307 | Solve for $q$, $ \dfrac{q + 3}{4q - 5} = -\dfrac{1}{4q - 5} + \dfrac{9}{4q - 5} $ | If we multiply both sides of the equation by $4q - 5$ , we get: $ q + 3 = -1 + 9$ $ q + 3 = 8$ $ q = 5 $ |
307 | Solve for $y$, $ \dfrac{9}{10y + 6} = \dfrac{2y - 9}{15y + 9} - \dfrac{3}{5y + 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $10y + 6$ $15y + 9$ and $5y + 3$ The common denominator is $30y + 18$ To get $30y + 18$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{9}{10y + 6} \times \dfrac{3}{3} = \dfrac... |
307 | Solve for $k$, $ \dfrac{6}{k} = -\dfrac{3k - 1}{k} + \dfrac{9}{k} $ | If we multiply both sides of the equation by $k$ , we get: $ 6 = -3k + 1 + 9$ $ 6 = -3k + 10$ $ -4 = -3k $ $ k = \dfrac{4}{3}$ |
307 | Solve for $z$, $ -\dfrac{z - 8}{20z^3} = -\dfrac{6}{16z^3} + \dfrac{8}{16z^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $20z^3$ $16z^3$ and $16z^3$ The common denominator is $80z^3$ To get $80z^3$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{z - 8}{20z^3} \times \dfrac{4}{4} = -\dfrac{4z - 3... |
307 | Solve for $q$, $ \dfrac{7}{8q + 8} = \dfrac{3q + 7}{6q + 6} + \dfrac{10}{2q + 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8q + 8$ $6q + 6$ and $2q + 2$ The common denominator is $24q + 24$ To get $24q + 24$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{7}{8q + 8} \times \dfrac{3}{3} = \dfrac{21... |
307 | Solve for $q$, $ -\dfrac{5q + 4}{3q} = -\dfrac{1}{15q} + \dfrac{4}{12q} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3q$ $15q$ and $12q$ The common denominator is $60q$ To get $60q$ in the denominator of the first term, multiply it by $\frac{20}{20}$ $ -\dfrac{5q + 4}{3q} \times \dfrac{20}{20} = -\dfrac{100q + 80}{60q}... |
307 | Solve for $y$, $ -\dfrac{8}{3y - 2} = \dfrac{4y + 5}{12y - 8} + \dfrac{5}{3y - 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3y - 2$ $12y - 8$ and $3y - 2$ The common denominator is $12y - 8$ To get $12y - 8$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{8}{3y - 2} \times \dfrac{4}{4} = -\dfrac{3... |
307 | Solve for $n$, $ \dfrac{2}{8n} = \dfrac{4n - 4}{4n} + \dfrac{1}{16n} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8n$ $4n$ and $16n$ The common denominator is $16n$ To get $16n$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{2}{8n} \times \dfrac{2}{2} = \dfrac{4}{16n} $ To get $16n$ in t... |
307 | Solve for $k$, $ \dfrac{8}{5k - 10} = -\dfrac{4}{k - 2} + \dfrac{k + 1}{4k - 8} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5k - 10$ $k - 2$ and $4k - 8$ The common denominator is $20k - 40$ To get $20k - 40$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{8}{5k - 10} \times \dfrac{4}{4} = \dfrac{3... |
307 | Solve for $r$, $ -\dfrac{r - 6}{15r^2} = -\dfrac{2}{3r^2} + \dfrac{3}{3r^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15r^2$ $3r^2$ and $3r^2$ The common denominator is $15r^2$ The denominator of the first term is already $15r^2$ , so we don't need to change it. To get $15r^2$ in the denominator of the second term, mult... |
307 | Solve for $k$, $ \dfrac{10}{k - 4} = \dfrac{4}{k - 4} - \dfrac{2k + 6}{k - 4} $ | If we multiply both sides of the equation by $k - 4$ , we get: $ 10 = 4 - 2k - 6$ $ 10 = -2k - 2$ $ 12 = -2k $ $ k = -6$ |
307 | Solve for $n$, $ \dfrac{2}{9n - 3} = -\dfrac{n + 1}{9n - 3} + \dfrac{5}{12n - 4} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $9n - 3$ $9n - 3$ and $12n - 4$ The common denominator is $36n - 12$ To get $36n - 12$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{2}{9n - 3} \times \dfrac{4}{4} = \dfrac{8... |
307 | Solve for $z$, $ -\dfrac{4z + 7}{5z + 1} = -\dfrac{6}{5z + 1} + \dfrac{6}{25z + 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5z + 1$ $5z + 1$ and $25z + 5$ The common denominator is $25z + 5$ To get $25z + 5$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{4z + 7}{5z + 1} \times \dfrac{5}{5} = -\df... |
307 | Solve for $z$, $ \dfrac{4z + 7}{3z + 6} = -\dfrac{9}{z + 2} - \dfrac{4}{5z + 10} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3z + 6$ $z + 2$ and $5z + 10$ The common denominator is $15z + 30$ To get $15z + 30$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{4z + 7}{3z + 6} \times \dfrac{5}{5} = \dfr... |
307 | Solve for $z$, $ \dfrac{3}{3z + 9} = \dfrac{10}{4z + 12} - \dfrac{z + 5}{5z + 15} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3z + 9$ $4z + 12$ and $5z + 15$ The common denominator is $60z + 180$ To get $60z + 180$ in the denominator of the first term, multiply it by $\frac{20}{20}$ $ \dfrac{3}{3z + 9} \times \dfrac{20}{20} = \... |
307 | Solve for $p$, $ \dfrac{8}{16p - 20} = -\dfrac{5p - 6}{4p - 5} - \dfrac{1}{20p - 25} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $16p - 20$ $4p - 5$ and $20p - 25$ The common denominator is $80p - 100$ To get $80p - 100$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{8}{16p - 20} \times \dfrac{5}{5} = \... |
307 | Solve for $x$, $ -\dfrac{8}{20x} = \dfrac{x - 6}{5x} + \dfrac{4}{20x} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $20x$ $5x$ and $20x$ The common denominator is $20x$ The denominator of the first term is already $20x$ , so we don't need to change it. To get $20x$ in the denominator of the second term, multiply it by ... |
307 | Solve for $t$, $ \dfrac{1}{16t^3} = \dfrac{t + 1}{4t^3} - \dfrac{5}{4t^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $16t^3$ $4t^3$ and $4t^3$ The common denominator is $16t^3$ The denominator of the first term is already $16t^3$ , so we don't need to change it. To get $16t^3$ in the denominator of the second term, mult... |
307 | Solve for $r$, $ \dfrac{4}{4r + 3} = -\dfrac{7}{16r + 12} - \dfrac{4r - 3}{16r + 12} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4r + 3$ $16r + 12$ and $16r + 12$ The common denominator is $16r + 12$ To get $16r + 12$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{4}{4r + 3} \times \dfrac{4}{4} = \dfra... |
307 | Solve for $n$, $ -\dfrac{10}{2n} = -\dfrac{10}{2n} - \dfrac{3n - 8}{2n} $ | If we multiply both sides of the equation by $2n$ , we get: $ -10 = -10 - 3n + 8$ $ -10 = -3n - 2$ $ -8 = -3n $ $ n = \dfrac{8}{3}$ |
307 | Solve for $p$, $ -\dfrac{7}{5p + 5} = -\dfrac{4p}{3p + 3} - \dfrac{3}{5p + 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5p + 5$ $3p + 3$ and $5p + 5$ The common denominator is $15p + 15$ To get $15p + 15$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ -\dfrac{7}{5p + 5} \times \dfrac{3}{3} = -\dfrac{... |
307 | Solve for $a$, $ -\dfrac{7}{25a^3} = -\dfrac{2a + 3}{20a^3} - \dfrac{3}{15a^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25a^3$ $20a^3$ and $15a^3$ The common denominator is $300a^3$ To get $300a^3$ in the denominator of the first term, multiply it by $\frac{12}{12}$ $ -\dfrac{7}{25a^3} \times \dfrac{12}{12} = -\dfrac{84}{... |
307 | Solve for $y$, $ \dfrac{8}{y - 1} = \dfrac{6}{4y - 4} + \dfrac{4y}{y - 1} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $y - 1$ $4y - 4$ and $y - 1$ The common denominator is $4y - 4$ To get $4y - 4$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{8}{y - 1} \times \dfrac{4}{4} = \dfrac{32}{4y - ... |
307 | Solve for $q$, $ \dfrac{4q - 3}{q^2} = \dfrac{10}{3q^2} + \dfrac{3}{3q^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $q^2$ $3q^2$ and $3q^2$ The common denominator is $3q^2$ To get $3q^2$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{4q - 3}{q^2} \times \dfrac{3}{3} = \dfrac{12q - 9}{3q^2} ... |
307 | Solve for $z$, $ \dfrac{8}{z - 4} = \dfrac{3z + 6}{3z - 12} - \dfrac{10}{5z - 20} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $z - 4$ $3z - 12$ and $5z - 20$ The common denominator is $15z - 60$ To get $15z - 60$ in the denominator of the first term, multiply it by $\frac{15}{15}$ $ \dfrac{8}{z - 4} \times \dfrac{15}{15} = \dfra... |
307 | Solve for $q$, $ -\dfrac{5}{4q - 2} = \dfrac{7}{4q - 2} + \dfrac{q + 1}{20q - 10} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4q - 2$ $4q - 2$ and $20q - 10$ The common denominator is $20q - 10$ To get $20q - 10$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{5}{4q - 2} \times \dfrac{5}{5} = -\dfra... |
307 | Solve for $k$, $ \dfrac{1}{15k - 15} = -\dfrac{1}{6k - 6} - \dfrac{4k + 4}{3k - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15k - 15$ $6k - 6$ and $3k - 3$ The common denominator is $30k - 30$ To get $30k - 30$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ \dfrac{1}{15k - 15} \times \dfrac{2}{2} = \dfra... |
307 | Solve for $a$, $ \dfrac{3}{16a + 16} = -\dfrac{3}{4a + 4} - \dfrac{4a - 5}{12a + 12} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $16a + 16$ $4a + 4$ and $12a + 12$ The common denominator is $48a + 48$ To get $48a + 48$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{3}{16a + 16} \times \dfrac{3}{3} = \df... |
307 | Solve for $k$, $ -\dfrac{8}{5k} = -\dfrac{5k + 10}{5k} - \dfrac{5}{4k} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5k$ $5k$ and $4k$ The common denominator is $20k$ To get $20k$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ -\dfrac{8}{5k} \times \dfrac{4}{4} = -\dfrac{32}{20k} $ To get $20k$ in... |
307 | Solve for $z$, $ \dfrac{4z + 7}{5z - 2} = \dfrac{4}{5z - 2} + \dfrac{7}{5z - 2} $ | If we multiply both sides of the equation by $5z - 2$ , we get: $ 4z + 7 = 4 + 7$ $ 4z + 7 = 11$ $ 4z = 4 $ $ z = 1$ |
307 | Solve for $n$, $ \dfrac{8}{5n + 1} = \dfrac{1}{15n + 3} + \dfrac{2n - 7}{25n + 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5n + 1$ $15n + 3$ and $25n + 5$ The common denominator is $75n + 15$ To get $75n + 15$ in the denominator of the first term, multiply it by $\frac{15}{15}$ $ \dfrac{8}{5n + 1} \times \dfrac{15}{15} = \df... |
307 | Solve for $x$, $ \dfrac{x + 8}{15x + 5} = -\dfrac{4}{3x + 1} - \dfrac{4}{15x + 5} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15x + 5$ $3x + 1$ and $15x + 5$ The common denominator is $15x + 5$ The denominator of the first term is already $15x + 5$ , so we don't need to change it. To get $15x + 5$ in the denominator of the seco... |
307 | Solve for $r$, $ -\dfrac{3}{r - 4} = \dfrac{7}{r - 4} + \dfrac{2r + 9}{5r - 20} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $r - 4$ $r - 4$ and $5r - 20$ The common denominator is $5r - 20$ To get $5r - 20$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{3}{r - 4} \times \dfrac{5}{5} = -\dfrac{15}{... |
307 | Solve for $q$, $ \dfrac{5}{4q^2} = \dfrac{10}{4q^2} + \dfrac{q - 8}{2q^2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4q^2$ $4q^2$ and $2q^2$ The common denominator is $4q^2$ The denominator of the first term is already $4q^2$ , so we don't need to change it. The denominator of the second term is already $4q^2$ , so we ... |
307 | Solve for $n$, $ -\dfrac{4n + 5}{15n + 10} = \dfrac{9}{9n + 6} - \dfrac{6}{12n + 8} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15n + 10$ $9n + 6$ and $12n + 8$ The common denominator is $180n + 120$ To get $180n + 120$ in the denominator of the first term, multiply it by $\frac{12}{12}$ $ -\dfrac{4n + 5}{15n + 10} \times \dfrac{... |
307 | Solve for $a$, $ -\dfrac{10}{2a - 4} = \dfrac{4}{10a - 20} - \dfrac{5a - 2}{2a - 4} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2a - 4$ $10a - 20$ and $2a - 4$ The common denominator is $10a - 20$ To get $10a - 20$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{10}{2a - 4} \times \dfrac{5}{5} = -\dfr... |
307 | Solve for $y$, $ \dfrac{10}{y - 3} = -\dfrac{9}{5y - 15} + \dfrac{2y - 4}{5y - 15} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $y - 3$ $5y - 15$ and $5y - 15$ The common denominator is $5y - 15$ To get $5y - 15$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{10}{y - 3} \times \dfrac{5}{5} = \dfrac{50}... |
307 | Solve for $p$, $ \dfrac{3p - 10}{10p} = \dfrac{6}{25p} - \dfrac{3}{5p} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $10p$ $25p$ and $5p$ The common denominator is $50p$ To get $50p$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{3p - 10}{10p} \times \dfrac{5}{5} = \dfrac{15p - 50}{50p} $ To... |
307 | Solve for $z$, $ \dfrac{4z + 1}{z} = -\dfrac{6}{3z} + \dfrac{6}{z} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $z$ $3z$ and $z$ The common denominator is $3z$ To get $3z$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{4z + 1}{z} \times \dfrac{3}{3} = \dfrac{12z + 3}{3z} $ The denominat... |
307 | Solve for $k$, $ \dfrac{6}{10k - 15} = -\dfrac{2}{8k - 12} - \dfrac{k - 9}{2k - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $10k - 15$ $8k - 12$ and $2k - 3$ The common denominator is $40k - 60$ To get $40k - 60$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{6}{10k - 15} \times \dfrac{4}{4} = \dfr... |
307 | Solve for $q$, $ \dfrac{10}{2q + 1} = -\dfrac{4}{2q + 1} - \dfrac{2q - 10}{2q + 1} $ | If we multiply both sides of the equation by $2q + 1$ , we get: $ 10 = -4 - 2q + 10$ $ 10 = -2q + 6$ $ 4 = -2q $ $ q = -2$ |
307 | Solve for $t$, $ -\dfrac{6}{25t + 15} = -\dfrac{4t + 10}{5t + 3} + \dfrac{5}{10t + 6} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25t + 15$ $5t + 3$ and $10t + 6$ The common denominator is $50t + 30$ To get $50t + 30$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{6}{25t + 15} \times \dfrac{2}{2} = -\d... |
307 | Solve for $n$, $ \dfrac{1}{4n - 4} = \dfrac{3}{4n - 4} - \dfrac{2n - 5}{4n - 4} $ | If we multiply both sides of the equation by $4n - 4$ , we get: $ 1 = 3 - 2n + 5$ $ 1 = -2n + 8$ $ -7 = -2n $ $ n = \dfrac{7}{2}$ |
307 | Solve for $n$, $ \dfrac{3n + 1}{2n} = \dfrac{10}{10n} + \dfrac{6}{10n} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2n$ $10n$ and $10n$ The common denominator is $10n$ To get $10n$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ \dfrac{3n + 1}{2n} \times \dfrac{5}{5} = \dfrac{15n + 5}{10n} $ The d... |
307 | Solve for $n$, $ \dfrac{6}{2n} = -\dfrac{5n + 1}{2n} - \dfrac{10}{2n} $ | If we multiply both sides of the equation by $2n$ , we get: $ 6 = -5n - 1 - 10$ $ 6 = -5n - 11$ $ 17 = -5n $ $ n = -\dfrac{17}{5}$ |
307 | Solve for $q$, $ -\dfrac{1}{3q - 3} = -\dfrac{10}{15q - 15} - \dfrac{q - 5}{3q - 3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3q - 3$ $15q - 15$ and $3q - 3$ The common denominator is $15q - 15$ To get $15q - 15$ in the denominator of the first term, multiply it by $\frac{5}{5}$ $ -\dfrac{1}{3q - 3} \times \dfrac{5}{5} = -\dfra... |
307 | Solve for $n$, $ \dfrac{3n}{9n - 6} = \dfrac{7}{9n - 6} - \dfrac{1}{3n - 2} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $9n - 6$ $9n - 6$ and $3n - 2$ The common denominator is $9n - 6$ The denominator of the first term is already $9n - 6$ , so we don't need to change it. The denominator of the second term is already $9n -... |
307 | Solve for $q$, $ \dfrac{9}{4q^3} = \dfrac{2q + 3}{q^3} - \dfrac{7}{q^3} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $4q^3$ $q^3$ and $q^3$ The common denominator is $4q^3$ The denominator of the first term is already $4q^3$ , so we don't need to change it. To get $4q^3$ in the denominator of the second term, multiply i... |
307 | Solve for $z$, $ \dfrac{5z - 5}{8z + 12} = \dfrac{10}{2z + 3} + \dfrac{8}{6z + 9} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8z + 12$ $2z + 3$ and $6z + 9$ The common denominator is $24z + 36$ To get $24z + 36$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{5z - 5}{8z + 12} \times \dfrac{3}{3} = \d... |
307 | Solve for $n$, $ -\dfrac{6}{3n} = -\dfrac{9}{12n} - \dfrac{4n - 3}{9n} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $3n$ $12n$ and $9n$ The common denominator is $36n$ To get $36n$ in the denominator of the first term, multiply it by $\frac{12}{12}$ $ -\dfrac{6}{3n} \times \dfrac{12}{12} = -\dfrac{72}{36n} $ To get $36... |
307 | Solve for $x$, $ \dfrac{1}{2x - 2} = \dfrac{2}{x - 1} + \dfrac{2x + 10}{x - 1} $ | First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2x - 2$ $x - 1$ and $x - 1$ The common denominator is $2x - 2$ The denominator of the first term is already $2x - 2$ , so we don't need to change it. To get $2x - 2$ in the denominator of the second term... |
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