Question

write an equation for a line passing through the point (2,5) that is parallel to ( y = \frac{2}{3}x + 2 ). then write a second equation for a line passing through the given point that is perpendicular to the given line. a slope - intercept equation for a line passing through the point (2,5) that is parallel to ( y=\frac{2}{3}x + 2 ) is (simplify your answer. type your answer in slope - intercept form. use integers or fractions for any numbers in the equation.)

Explanation:

Part 1: Equation of the parallel line

Step 1: Recall the slope of parallel lines

Parallel lines have the same slope. The given line is \( y = \frac{2}{3}x + 2 \), so its slope \( m \) is \( \frac{2}{3} \).

Step 2: Use the point - slope form \( y - y_1=m(x - x_1) \)

We have the point \( (x_1,y_1)=(2,5) \) and \( m=\frac{2}{3} \). Substitute these values into the point - slope form:
\( y - 5=\frac{2}{3}(x - 2) \)

Step 3: Convert to slope - intercept form (\( y=mx + b \))

Expand the right - hand side:
\( y - 5=\frac{2}{3}x-\frac{4}{3} \)
Add 5 to both sides. Since \( 5=\frac{15}{3} \), we have:
\( y=\frac{2}{3}x-\frac{4}{3}+\frac{15}{3} \)
\( y=\frac{2}{3}x+\frac{11}{3} \)

Part 2: Equation of the perpendicular line

Step 1: Recall the slope of perpendicular lines

If two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other line. The slope of the given line is \( \frac{2}{3} \), so the slope of the perpendicular line \( m_{\perp}=-\frac{3}{2} \)

Step 2: Use the point - slope form \( y - y_1=m(x - x_1) \)

We have the point \( (x_1,y_1)=(2,5) \) and \( m =-\frac{3}{2} \). Substitute these values into the point - slope form:
\( y - 5=-\frac{3}{2}(x - 2) \)

Step 3: Convert to slope - intercept form (\( y=mx + b \))

Expand the right - hand side:
\( y - 5=-\frac{3}{2}x + 3 \)
Add 5 to both sides. Since \( 5 = \frac{10}{2} \), we have:
\( y=-\frac{3}{2}x+3 + 5 \)
\( y=-\frac{3}{2}x + 8 \)

Answer:

• Equation of the line parallel to \( y=\frac{2}{3}x + 2 \) and passing through \( (2,5) \): \( y=\frac{2}{3}x+\frac{11}{3} \)
• Equation of the line perpendicular to \( y=\frac{2}{3}x + 2 \) and passing through \( (2,5) \): \( y=-\frac{3}{2}x + 8 \)