Question
Consider the line that passes through the point \( \mathrm{P}(4,5) \) and has slope -2 . Write the slope-intercept form of the equation of the line. Use exact values.
Ask by Rodriguez Weaver. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equation of the line in slope-intercept form is \( y = -2x + 13 \).
Solution
To find the slope-intercept form of the equation of a line, we can use the formula:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
Given:
- The slope \( m = -2 \)
- The point \( P(4, 5) \)
We can substitute the slope and the coordinates of point \( P \) into the equation to find \( b \).
1. Substitute \( m \), \( x \), and \( y \) into the equation:
\[
5 = -2(4) + b
\]
2. Calculate \( -2(4) \):
\[
-2(4) = -8
\]
3. Substitute this value back into the equation:
\[
5 = -8 + b
\]
4. Solve for \( b \):
\[
b = 5 + 8 = 13
\]
Now we have both \( m \) and \( b \):
- Slope \( m = -2 \)
- Y-intercept \( b = 13 \)
Thus, the slope-intercept form of the equation of the line is:
\[
y = -2x + 13
\]
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Simplify this solution Mind Expander
To find the slope-intercept form of the equation of the line, we start with the point-slope form, which is given by \( y - y_1 = m(x - x_1) \). Here, the slope \( m = -2 \) and the point \( P(4, 5) \) gives us \( x_1 = 4 \) and \( y_1 = 5 \). Substituting these values into the point-slope formula: \[ y - 5 = -2(x - 4) \] Now, simplify this equation: \[ y - 5 = -2x + 8 \] \[ y = -2x + 8 + 5 \] \[ y = -2x + 13 \] Thus, the slope-intercept form of the equation of the line is \( y = -2x + 13 \).
