Questions: Write the slope-intercept form of an equation of the line that passes through the given point and is parallel to the graph of each equation. 5. (3,2), y=x+5 6. (-2,5), y=-4x+2

Write the slope-intercept form of an equation of the line that passes through the given point and is parallel to the graph of each equation.
5. (3,2), y=x+5
6. (-2,5), y=-4x+2

Transcript text: Write the slope-intercept form of an equation of the line that passes through the given point and is parallel to the graph of each equation. 5. $(3,2), y=x+5$ 6. $(-2,5), y=-4 x+2$

Solution

Solution Steps

Step 1: Identify the Slope of the Given Lines

For the first line $ y = x + 5 $, the slope $ m_1 $ is $ 1 $.
For the second line $ y = -4x + 2 $, the slope $ m_2 $ is $ -4 $.

Step 2: Use the Point-Slope Form for the First Line

Using the point $ (3, 2) $ and the slope $ m_1 = 1 $, we apply the point-slope form: \[ y - y_1 = m(x - x_1) \] Substituting the values: \[ y - 2 = 1(x - 3) \]

Step 3: Convert to Slope-Intercept Form for the First Line

Rearranging the equation: \[ y - 2 = x - 3 \implies y = x - 1 \] Thus, the slope-intercept form for the first line is $ y = 1x - 1 $.

Step 4: Use the Point-Slope Form for the Second Line

Using the point $ (-2, 5) $ and the slope $ m_2 = -4 $, we apply the point-slope form: \[ y - y_1 = m(x - x_1) \] Substituting the values: \[ y - 5 = -4(x + 2) \]

Step 5: Convert to Slope-Intercept Form for the Second Line

Rearranging the equation: \[ y - 5 = -4x - 8 \implies y = -4x - 3 \] Thus, the slope-intercept form for the second line is $ y = -4x - 3 $.