Question

write the slope, point - slope form, and slope - intercept form of the equation of the line described through: (1, 3), parallel to y = 7x - 2

Explanation:

Step1: Determine the slope

Parallel lines have the same slope. The given line is \( y = 7x - 2 \), which is in slope - intercept form \( y=mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept). So the slope \( m \) of the given line is 7. Therefore, the slope of the line we want to find is also \( m = 7 \).

Step2: Point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1) \) is a point on the line and \( m \) is the slope. We know that the line passes through the point \( (1,3) \) (so \( x_1 = 1 \) and \( y_1=3 \)) and has a slope \( m = 7 \). Substituting these values into the point - slope formula, we get:
\( y - 3=7(x - 1) \)

Step3: Slope - intercept form

We start with the point - slope form \( y - 3=7(x - 1) \). First, distribute the 7 on the right - hand side: \( y-3 = 7x-7 \). Then, add 3 to both sides of the equation to solve for \( y \): \( y=7x-7 + 3 \), which simplifies to \( y=7x-4 \)

Answer:

• Slope: \( 7 \)
• Point - slope form: \( y - 3=7(x - 1) \)
• Slope - intercept form: \( y = 7x-4 \)