rewards sequencelengths 8 8 | prompt stringlengths 236 4.77k | gt stringlengths 1 1.1k | answers sequencelengths 8 8 |
|---|---|---|---|
[
1,
1,
1,
1,
-0.5,
1,
1,
1
] | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the equation: $\frac{x}{{x-1}}=\frac{{x-3}}{{2x-2}}$. Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_start|>assistant
| x=-3 | [
"Let's solve this equation step by step. So, we have:\n\n\\[\n\\frac{x}{x-1} = \\frac{x-3}{2x-2}\n\\]\n\nFirst off, I notice that the denominator on the right side, 2x - 2, can be factored. Factoring it gives us 2(x - 1). So, let's rewrite the equation:\n\n\\[\n\\frac{x}{x-1} = \\frac{x-3}{2(x-1)}\n\\]\n\nNow, I se... |
[
1,
1,
1,
1,
1,
1,
1,
1
] | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a new ellipse, the center is at $(-4,2)$, one focus is at $(-4,0)$, and one endpoint of a semi-major axis is at $(-4,5)$. Find the semi-minor axis of this ellipse. Let's think step by step and output ... | \sqrt{5} | [
"Let's tackle this ellipse problem step by step. So, we have an ellipse with a center at (-4,2), one focus at (-4,0), and one endpoint of the semi-major axis at (-4,5). We need to find the length of the semi-minor axis.\n\nFirst, I recall that in an ellipse, the center is the midpoint between the two foci, and it's... |
[
-1,
-0.5,
-1,
-1,
-0.5,
-1,
-0.5,
-1
] | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the coordinate plane, the curve $xy = 2$ intersects a circle at four points, three of which are $\left( 4, \frac{1}{2} \right),$ $\left( -2, -1 \right),$ and $\left( \frac{2}{3}, 3 \right).$ Find the ... | \left(-\frac{3}{4}, -\frac{8}{3}\right) | [
"So I've got this problem here. There's this curve defined by the equation \\( xy = 2 \\), and it intersects with a circle at four points. Three of those points are given: \\( (4, \\frac{1}{2}) \\), \\( (-2, -1) \\), and \\( (\\frac{2}{3}, 3) \\). I need to find the fourth point of intersection.\n\nFirst off, since... |
[
-0.5,
-0.5,
-0.5,
-0.5,
-0.5,
-0.5,
-0.5,
-0.5
] | <|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A line through the points $(1, -5)$ and $(k, 23)$ is parallel to the line $4x + 6y = 30$. What is the value of $k$? Let's think step by step and output the final answer within \boxed{}<|im_end|>
<|im_sta... | 43 | [
"So I've got this problem here. I need to find the value of \\(k\\) such that the line passing through the points \\((1, -5)\\) and \\((k, 23)\\) is parallel to the line given by the equation \\(4x + 6y = 30\\). Okay, let's break this down step by step.\n\nFirst, I need to recall that parallel lines have the same s... |
[
-1,
1,
1,
1,
1,
1,
1,
-1
] | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | (-8,0] | ["So I have this problem: I need to determine the range of real number \\( a \\) such that the propo(...TRUNCATED) |
[
1,
1,
-0.5,
1,
1,
1,
-0.5,
1
] | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | 50 | ["So I've got this problem here about three dice stacked up, and only five of the eighteen faces are(...TRUNCATED) |
[
1,
1,
-1,
1,
-0.5,
-0.5,
1,
1
] | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | 240 | ["I've got this problem here: compute \\(\\sqrt{128} \\cdot \\sqrt{50} \\cdot \\sqrt[3]{27}\\). At f(...TRUNCATED) |
[
-0.5,
1,
1,
1,
1,
1,
-0.5,
1
] | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | -1 | ["I have this problem about a prism with 21 edges, x faces, and y vertices. I need to find the value(...TRUNCATED) |
[
1,
-1,
1,
1,
-0.5,
-0.5,
-0.5,
1
] | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | [-\frac{8}{3}, 10 + 4\sqrt{2}) | ["Let's tackle this problem step by step. So, we have an ellipse given by the equation:\n\n\\[\n\\fr(...TRUNCATED) |
[
1,
1,
1,
-0.5,
-0.5,
1,
1,
1
] | "<|im_start|>system\nPlease reason step by step, and put your final answer within \\boxed{}.<|im_end(...TRUNCATED) | 4 | ["So here's this problem about an arithmetic sequence. Let's see what it's asking.\n\nWe have an ari(...TRUNCATED) |
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