Question

1. write an equation in slope-intercept form for the line that passes through (0, 6) an is parallel to the line described by $y = 2x + 3$.
2. write an equation in slope-intercept form for the line that passes through (4, 6) an is perpendicular to the line described by $y = x - 3$.

Explanation:

Problem 15 (Equation of Parallel Line)

Step1: Recall Slope-Intercept Form

Slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step2: Determine Slope of Parallel Line

Parallel lines have equal slopes. The given line is \( y = 2x + 3 \), so its slope \( m = 2 \). Thus, the new line also has \( m = 2 \).

Step3: Find Y-Intercept

The line passes through \( (0, 6) \). In slope-intercept form, when \( x = 0 \), \( y = b \). So \( b = 6 \).

Step4: Write the Equation

Substitute \( m = 2 \) and \( b = 6 \) into \( y = mx + b \), we get \( y = 2x + 6 \).

Step1: Recall Slope of Perpendicular Lines

If two lines are perpendicular, the product of their slopes is \( -1 \). The given line is \( y = x - 3 \), so its slope \( m_1 = 1 \). Let the slope of the perpendicular line be \( m_2 \). Then \( m_1 \times m_2 = -1 \), so \( 1 \times m_2 = -1 \), which gives \( m_2 = -1 \).

Step2: Use Point-Slope Form to Find Equation

Point-slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (4, 6) \) and \( m = -1 \). Substitute these values: \( y - 6 = -1(x - 4) \).

Step3: Simplify to Slope-Intercept Form

Expand the right side: \( y - 6 = -x + 4 \). Add 6 to both sides: \( y = -x + 4 + 6 \), so \( y = -x + 10 \).

Answer:

\( y = 2x + 6 \)

Problem 16 (Equation of Perpendicular Line)