Questions: Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the equation. (0,2); y=-5x+8

Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the equation. (0,2); y=-5x+8
Transcript text: Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the equation. \[ (0,2) ; y=-5 x+8 \]
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Solution

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Solution Steps

Step 1: Identify the Slope of the Given Line

The equation of the given line is \( y = -5x + 8 \). From this equation, we can see that the slope \( m_{\text{given}} \) is \( -5 \).

Step 2: Determine the Slope of the Perpendicular Line

The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope \( m_{\text{perpendicular}} \) is given by: \[ m_{\text{perpendicular}} = -\frac{1}{m_{\text{given}}} = -\frac{1}{-5} = \frac{1}{5} \]

Step 3: Use the Point-Slope Form

We have the point \( (0, 2) \) through which the line passes. Using the point-slope form of the equation: \[ y - y_1 = m(x - x_1) \] Substituting \( (x_1, y_1) = (0, 2) \) and \( m = \frac{1}{5} \): \[ y - 2 = \frac{1}{5}(x - 0) \]

Step 4: Convert to Slope-Intercept Form

To convert the equation to slope-intercept form \( y = mx + b \), we simplify: \[ y - 2 = \frac{1}{5}x \] Adding \( 2 \) to both sides gives: \[ y = \frac{1}{5}x + 2 \]

Thus, the equation of the line in slope-intercept form that passes through the point \( (0, 2) \) and is perpendicular to the line \( y = -5x + 8 \) is: \[ y = \frac{1}{5}x + 2 \]

Final Answer

\(\boxed{y = \frac{1}{5}x + 2}\)

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