Question
A line passes through the point \( (-4,-7) \) and has a slope of \( -\frac{5}{4} \). Write an equation in slope-intercept form for this line.
Ask by Parry Grant. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equation of the line in slope-intercept form is \( y = -\frac{5}{4}x - 12 \).
Solution
To find the equation of a line in **slope-intercept form**, which is:
\[
y = mx + b
\]
where:
- \( m \) is the slope of the line
- \( b \) is the y-intercept (the point where the line crosses the y-axis)
**Given:**
- Slope (\( m \)) = \( -\frac{5}{4} \)
- A point on the line = \( (-4, -7) \)
**Steps to Find the Equation:**
1. **Plug the known values into the slope-intercept form.**
We know \( m = -\frac{5}{4} \) and the line passes through \( (-4, -7) \), so:
\[
-7 = \left(-\frac{5}{4}\right)(-4) + b
\]
2. **Solve for \( b \) (the y-intercept).**
\[
-7 = \left(-\frac{5}{4}\right) \times (-4) + b
\]
\[
-7 = 5 + b
\]
\[
b = -7 - 5
\]
\[
b = -12
\]
3. **Write the final equation with the slope and y-intercept.**
\[
y = -\frac{5}{4}x - 12
\]
**Final Answer:**
\[
\boxed{y = -\dfrac{5}{4}\,x\ -\ 12}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find the equation of a line in slope-intercept form \( y = mx + b \), we start with the slope \( m = -\frac{5}{4} \) and the point \( (-4, -7) \). We can use the point-slope form of the equation first, which is given by \( y - y_1 = m(x - x_1) \). Substituting in the values, we have: \[ y - (-7) = -\frac{5}{4}(x - (-4)) \] This simplifies to: \[ y + 7 = -\frac{5}{4}(x + 4) \] Now, distribute the slope: \[ y + 7 = -\frac{5}{4}x - 5 \] Next, isolate \( y \) by subtracting 7 from both sides: \[ y = -\frac{5}{4}x - 5 - 7 \] Finally, combine the constant terms: \[ y = -\frac{5}{4}x - 12 \] So, the equation of the line in slope-intercept form is: \[ y = -\frac{5}{4}x - 12 \]
