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Wize High School Grade 10 Math Textbook > Linear Relations (Review)

Linear Equations

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Linear Equations

A linear relation can be represented by a linear equation, which is an equation with a degree of 1.

3 Different Forms of a Linear Equation

  1. Slope y-intercept form

  1. Standard form

  1. Point-slope form


Wize Tip
If you are given two points on the line and have to find the slope, use any one of these formulas:
m=riserun\large\boxed{m=\dfrac{\text{rise}}{\text{run}}} or m=change in ychange in x\large\boxed{m=\dfrac{\text{change in y}}{\text{change in x}}} or m=y2−y1x2−x1\large\boxed{m=\dfrac{y_2-y_1}{x_2-x_1}}

Practice: Converting Between Linear Equations

Answer the questions that follow each of the following linear equations.
y=−5x+3y=-5x+3
a) Rewrite this equation in standard form.
b) Identify the slope of the line.
c) Identify the y-intercept of the line.
d) True or False. The point (1,−2)(1,-2) is on this line.
e) True or False. The point (0,0)(0, 0) is on this line.

Practice: Finding Equations of Lines

Determine the equation of each line based on the information given.
a) The line that has a slope of 33 and passes through the point (0,−1)(0,-1).

b) The line that has a slope of −2-2 and passes through the point (−1,3)(-1,3).

c) The line that passes through the point (3,−8)(3,-8) and (1,−6)(1, -6).

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Parallel & Perpendicular Lines


When we are given the equation of two lines, we can determine if the lines are parallel or perpendicular by looking at the slopes of the lines.

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Parallel Lines

Two lines are parallel if they have the same slope.

(If two lines have the same slope, then they are parallel)

Example
Show that the lines y=−2x+3y=-2x+3 and 4x+2y=04x+2y=0 are parallel.

The line y=−2x+3y=-2x+3 has a slope m=−2m=-2.

Rearranging the line 4x+2y=04x+2y=0 into slope y-intercept form:
4x+2y=0−4x         −4x 2y=−4x 2  =  2 y=−2x\begin{array}{rcl} \cancel{4x}+2y&=&0\\ \colorFour{\cancel{-4x}}~~~~~~~~~&&\colorFour{-4x}\\\\ \cancel{~2}y&=&-4x\\ \colorFour{\dfrac{}{\cancel{~2}}}~~&=&~\colorFour{\dfrac{}{~2~}}\\\\ y&=&-2x \end{array}
We see that this line also has a slope of m=−2m=-2.

Therefore, the lines are parallel.
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Perpendicular Lines

Two lines are perpendicular if their slopes are negative reciprocals of one another.

(If two lines have negative reciprocal slopes, then they are perpendicular)


Wize Tip
The negative reciprocal of a number is just the "negative flip" of that number.

Examples
  • 35\dfrac{3}{5} and −53-\dfrac{5}{3} are negative reciprocals of one another
  • −2-2 and 12\dfrac{1}{2} are negative reciprocals of one another
  • −0.25-0.25 and 44 are negative reciprocals of one another

Example
Write the equation of a line that is perpendicular to y=4−3xy=4-3x.

The slope of the line y=4−3xy=4-3x is −3-3. The negative reciprocal of this number is 13\dfrac{1}{3}.

So, any line that perpendicular to y=4−3xy=4-3x must have a slope of 13\dfrac{1}{3}. One example would by y=13xy=\dfrac{1}{3}x.

There are many possible answers because there are many lines that are perpendicular to y=4−3xy=4-3x, we just have to change the y-intercept value and we have another line that is perpendicular to the given line.

Practice: Finding Equations of Lines

Determine the equation of each line based on the information given.
a) The line that passes through the point (4,4)(4,4) and is parallel to the line x−2y−8=0x-2y-8=0.

b) The line that passes through the point (0,0)(0,0) and is perpendicular to the line 2x+3y−5=02x+3y-5=0.