Question

write an equation of a line in slope-intercept form that passes through (1.4, -3) and is perpendicular to the line that passes through (-4, 7) and (9, 8).
\\( y = 13x - 15.2 \\)
\\( y = \frac{1}{13}x + 15.2 \\)
\\( y = -\frac{1}{13}x - 15.2 \\)
\\( y = -13x + 15.2 \\)

Explanation:

Step1: Calculate the slope of the line passing through \((-4,7)\) and \((9,8)\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substitute \(x_1=-4,y_1 = 7,x_2=9,y_2 = 8\) into the formula:
\(m=\frac{8 - 7}{9-(-4)}=\frac{1}{13}\)

Step2: Find the slope of the perpendicular line

If two lines with slopes \(m_1\) and \(m_2\) are perpendicular, then \(m_1\times m_2=-1\).
Let \(m_1=\frac{1}{13}\), then \(m_2=-13\)

Step3: Use the point - slope form \(y - y_0=m(x - x_0)\) to find the equation of the line

We know \(m=-13\) and the point \((x_0,y_0)=(1.4,-3)\)
\(y-(-3)=-13(x - 1.4)\)
\(y + 3=-13x+18.2\)

Step4: Convert to slope - intercept form \(y=mx + b\)

Subtract 3 from both sides of the equation:
\(y=-13x+18.2-3\)
\(y=-13x + 15.2\)

Answer:

\(y=-13x + 15.2\)