Question
Find the equation of the line using the point-slope formula. Write the final equation using the slope-intercept form. \[ (x, y)=(-2,10) \text { and }(x, y)=(5,-4) \text { are points on the line } \]
Ask by Todd Phillips. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equation of the line in slope-intercept form is \( y = -2x + 6 \).
Solution
1. **Calculate the slope \( m \)**
Using the two given points \((-2, 10)\) and \((5, -4)\), the slope is
\[
m = \frac{-4 - 10}{5 - (-2)} = \frac{-14}{7} = -2.
\]
2. **Write the point-slope form**
Choose the point \((-2, 10)\) for example, and substitute the slope \( m = -2 \) into
\[
y - y_1 = m(x - x_1):
\]
\[
y - 10 = -2(x - (-2)) = -2(x + 2).
\]
3. **Convert to slope-intercept form**
Distribute and simplify:
\[
y - 10 = -2x - 4.
\]
Add 10 to both sides to isolate \( y \):
\[
y = -2x - 4 + 10,
\]
\[
y = -2x + 6.
\]
The final equation of the line in slope-intercept form is
\[
y = -2x + 6.
\]
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Simplify this solution The Deep Dive
To find the equation of the line using the point-slope formula, we first need to determine the slope \(m\) using the two points \((-2, 10)\) and \((5, -4)\). The slope is calculated as follows: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 10}{5 - (-2)} = \frac{-14}{7} = -2. \] Now, using the point-slope formula \(y - y_1 = m(x - x_1)\) with point \((-2, 10)\): \[ y - 10 = -2(x + 2). \] Distributing the slope on the right side, we get: \[ y - 10 = -2x - 4. \] Adding 10 to both sides results in: \[ y = -2x + 6. \] So the final equation in slope-intercept form is: \[ y = -2x + 6. \]
