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1. DO YOU KNOW THE LINE?
Tackle the Math
All Things Lines

2. Example of Linear Equation
■ Linear equations are always straight lines when graphed.
■ They always fit the following formulas:
– Slope-intercept form: 𝑦 = 𝑚𝑥 + 𝑏, where m
is the slope and b is the y-intercept.
– Point-slope form: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1),
where m is the slope and 𝑥1, 𝑦1 is a point.
– General form: 𝐴𝑥 + 𝐵𝑦 = 𝐶
■ Slope-intercept form and point-slope form are by far the
most common.

3. Slope
■ The “slope” of a line is the measure of its steepness.
■ Other terms for slope are average rate of change and rise over run
■ The equation for slope is as follows:
– 𝑚 =
𝑦2−𝑦1
𝑥2−𝑥1
– To find the slope, you must have two points 𝑥1, 𝑦1 𝑎𝑛𝑑 𝑥2, 𝑦2 .

4. Slope Example
Find the slope of a line with points (1, 3) and (4, 12).
■ First, determine 𝑥1, 𝑦1 𝑎𝑛𝑑 𝑥2, 𝑦2 .
■ 𝑥1 = 1, 𝑦1 = 3, 𝑥2 = 4, 𝑎𝑛𝑑 𝑦2 = 12.
■ Now let’s use the slope equation.
■ 𝑚 =
𝑦2−𝑦1
𝑥2−𝑥1
■ 𝑚 =
12−3
4−1
■ 𝑚 =
9
3
■ 𝑚 = 3

5. Equation Example
Find the equation of a line which has a slope of 2 and goes through the point (1, 5).
■ Since we already know the slope, it would be prudent to use slope-intercept form:
𝑦 = 𝑚𝑥 + 𝑏. m = 2.We do not, however, know the y-intercept (b).This is not a
problem.We can replace m, x, and y with the slope and point given.
■ 𝑦 = 𝑚𝑥 + 𝑏
■ 5 = 2 1 + 𝑏. Now we solve for b.
■ 5 = 2 + 𝑏
■ 3 = 𝑏. Now we have everything we need! Let’s plug in m and b to slope-intercept
form.
■ 𝑦 = 2𝑥 + 3

6. Equation Example 2
■ 𝑚 =
8−0
2−(−2)
=
8
4
= 2
■ 𝑦 = 2𝑥 + 𝑏
■ 0 = 2 −2 + 𝑏
■ 0 = −4 + 𝑏
■ 4 = 𝑏
■ 𝑦 = 2𝑥 + 4
Find the equation of the line passing through the points (-2, 0) and (2, 8).
Slope-Intercept
𝑦 = 𝑚𝑥 + 𝑏
Point-Slope
𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1
■ 𝑚 =
8−0
2−(−2)
=
8
4
= 2
■ 𝑦 − 𝑦1 = 2 𝑥 − 𝑥1
■ 𝑦 − 8 = 2 𝑥 − 2
■ 𝑦 − 8 = 2𝑥 − 4
■ 𝑦 = 2𝑥 + 4

7. Parallel Lines
■ Lines are considered parallel when they
never touch.
■ Algebraically, they have the same slope.
■ If you’re given a problem which says, “Find
the equation of a line passing through the
point (1, 2) and parallel to 𝑦 = 3𝑥 − 4, you
know the new slope will be 3

8. Perpendicular Lines
■ Lines are considered perpendicular when
they cross one another at a 90˚ angle.
■ Algebraically, if one line has a slope of 𝑚, a
line perpendicular to it will have a slope of
−
1
𝑚
. (Opposite reciprocal)
■ The green line in the picture has a slope of
2
3
.
The blue line, which is perpendicular to it,
has a slope of −
3
2
.

9. Equation Example 3
Since parallel lines have the same
slope, we need to find the slope of
the given line.
■ 6x + 2y = 32
■ 2y = −6x + 32
■ y = −3x + 16
■ 𝑚 = −3
Find the equation of the line parallel to the line 6𝑥 + 2𝑦 = 32 and goes
through the point (2,7).
Now we can find the equation.
■ y − 7 = −3(x − 2)
■ y − 7 = −3x + 6
■ y = −3x + 13

10. Your Turn
Find the equation of the line perpendicular to the line 2𝑥 + 4𝑦 = 16 and goes
through the point (2,7).
Find the equation of the line that passes through the points (−3,7) and 5, −9 .

11. UPCOMING
EVENTS
■ Tackle the Math Series
– Why Does Order Matter?
– 4 Out of 3 People Struggle
with Math
– Probably Probability
– Do You Know the Line?
– Beating the System (of
Equations)

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