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Wize High School Grade 12 Calculus Textbook > Review

Linear Equations (Lines)

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Equations of Lines

Here are two common forms of the equation of a line:

1. y=mx+b\orange{y=mx+b}
  • m=riserun=change in ychange in x\displaystyle m=\frac{\text{rise}}{\text{run}}=\frac{\text{change in}\ y}{\text{change in}\ x} is the slope of the line
  • The point (0, b)\left(0,\ b\right) is the y-intercept of the line
2. y−y0=m(x−x0)\orange {y-y_0=m(x-x_0)}
  • m=riserun=change in ychange in x\displaystyle m=\frac{\text{rise}}{\text{run}}=\frac{\text{change in}\ y}{\text{change in}\ x} is the slope of the line
  • The point (x0,y0)(x_0, y_0) is any point on the line

Wize Tip
A normal line to the line y=mx+by=mx+b is a line that is perpendicular to y=mx+by=mx+b → it has slope −1m-\frac{1}{m}


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Example
Find the equation of the line that passes through the point (1, 0)\left(1,\ 0\right) and is
a) parallel to 3x−4y=53x-4y=5
b) perpendicular to 3x−4y=53x-4y=5

Rearrange the equation:
3x−4y=53x-4y=5
−4y=−3x+5-4y=-3x+5
y=−3−4x+5−4y=\frac{-3}{-4}x+\frac{5}{-4}
y=34x−54y=\frac{3}{4}x-\frac{5}{4}
This line has slope m=34m=\frac{3}{4}

a) The slope of a parallel line is 34\frac{3}{4}.
Therefore, the equation of the line is:
y−0=34(x−1)y-0=\frac{3}{4}\left(x-1\right)
y=34x−34y=\frac{3}{4}x-\frac{3}{4}

b) The slope of perpendicular line is −43-\frac{4}{3} (negative reciprocal).
Therefore, the equation of the line is:
y−0=−43(x−1)y-0=-\frac{4}{3}\left(x-1\right)
y=−43x+43y=-\frac{4}{3}x+\frac{4}{3}

Practice: Equations of Lines

Find the equation of the line in y=mx+by=mx+b form that passes through the points (1,− 1)\left(1,-\ 1\right) and (−2,−3)(-2, -3).