Question

finding the equation of a line
ex. find the equation of each line.
a)
(grid image)

b) perpendicular to $y = -\frac{2}{3}x - 8$, and passes through (4, 5)

c) passes through the points (5, 9) and (8, 18)

finding the equation of a line
ex: graph the following lines
on the same plane. state their
point of intersection
$y = -\frac{3}{2}x + 6$
$y = 2x - 1$
(grid image)

Explanation:

Part (a)

Step1: Identify slope (m)

The line is horizontal, so rise (change in y) is 0. Run (change in x) is non - zero. So \(m=\frac{\text{rise}}{\text{run}}=\frac{0}{1} = 0\) (any non - zero run, here we take run = 1 for simplicity).

Step2: Identify y - intercept (b)

The line crosses the y - axis at \(y = 2\), so \(b = 2\).

Step3: Use slope - intercept form \(y=mx + b\)

Substitute \(m = 0\) and \(b = 2\) into the formula: \(y=0\times x+2\), which simplifies to \(y = 2\).

Step1: Find the slope of the perpendicular line

The slope of the given line \(y=-\frac{2}{3}x - 8\) is \(m_1=-\frac{2}{3}\). The slope of a line perpendicular to a line with slope \(m_1\) is \(m_2=-\frac{1}{m_1}\). So \(m_2=-\frac{1}{-\frac{2}{3}}=\frac{3}{2}\).

Step2: Use point - slope form \(y - y_1=m(x - x_1)\)

We have the point \((x_1,y_1)=(4,5)\) and \(m=\frac{3}{2}\). Substitute into the formula: \(y - 5=\frac{3}{2}(x - 4)\).

Step3: Simplify to slope - intercept form

\(y-5=\frac{3}{2}x-6\), then \(y=\frac{3}{2}x-6 + 5=\frac{3}{2}x-1\).

Step1: Calculate the slope (m)

Using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with \((x_1,y_1)=(5,9)\) and \((x_2,y_2)=(8,18)\). So \(m=\frac{18 - 9}{8 - 5}=\frac{9}{3}=3\).

Step2: Use point - slope form

Using the point \((5,9)\) and \(m = 3\), the point - slope form is \(y - 9=3(x - 5)\).

Step3: Simplify to slope - intercept form

\(y-9 = 3x-15\), then \(y=3x-15 + 9=3x - 6\).

Answer:

\(y = 2\)

Part (b)