Linear Equations
Objectives: Find slope
Graph Lines
Write equation in slope-intercept and general
form
Slope
 The slant of a line is called its slope. Slope is
measured by its rise as compared to its run.
Rise / Run
 Rise is the vertical change in the line
 Run is the horizontal change in the line
 Mathematically you can find rise by
subtracting the y’s
 Mathematically you can find run by
subtracting the x’s.
Slope Formula: m = (y2 – y1) / (x2 - x1)
 Find the slope of the line that passes through
the points (2/5, -1/2) and (3/4, 1/3)
 (-1/2 – 1/3) / (2/5 – ¾) need common
denominator
 (-3/6 – 2/6) / (8/20 – 15/20)
 (-5/6) / (-7/20)
when dividing
fractions invert and -5/6
x -20/7 = 100/42
multiply
 50/21
reduce
 When calculating slope if the rise is 0, called
‘0 slope’, the line is horizontal (No RISE).
 When calculating slope if the run is 0, called
‘no slope’, the line is vertical (NO RUN)
 When the slope of a line is positive, the line
slants up from left to right.
 When the slope of a line is negative – the line
slants down from left to right.
 When the slope of a line is 0 – the line is
horizontal.
 When the slope of a line is ‘no slope’ – the
line is vertical.
Find the slope of the line passing
through
 (1,2) and ((5, -3)
 (-3 – 2) / (5 – 1) => -5 / 4
 (2/3, -4) and (2/3, -2)
 ( -2 - -4) / (2/3 – 2/3)
=> 2 / 0 = no slope
Slope-Intercept Form: y = mx + b
 In slope intercept form the number with the x (m) is
the slope.
 The number by itself it the y-intercept.
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Find the slope and y-intercept of 3x + 2y = 6
Get y by itself:
2y = 6 – 3x
Subtract 3x from both sides
Y = 3 – 3/2 x
Divide both sides by 2
Slope: - 3/2
Y-intercept: 3
Graph the line: 3x – 2y = 8
 Get y by itself:
-2y = 8 – 3x
y = -4 + 3/2 x
 Plot y-intercept (this is the number by itself):
Mark the point -4 on the y-axis (down 4)
 Count slope (this is the number with the x –
numerator is rise, denominator is run) from this
point:
From this point go up 3 and right 2 and plot a 2nd
point (slope of 3/2)
 Connect
Graph the line that contains the point
(5, -3) and has a slope of -4/5
 Mark the point (5, -3) on the coordinate plane.
Right 5 and down 3
 From this point count your slope (-4/5)
Down 4 and right 5
 Connect the two points
Graph the line: x = 5
 When there is only an x the value of x for all points
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have to be that number.
For example
(5,2)
(5,-3)
(5,0)
Connect these points. What do you find?
When only an x in the equation the graph is a vertical
line through that value.
Graph: y = -2
 In this case the y values always have to be
the same.
 (4, -2)
 (-3, -2)
 (0, -2)
 Graph and connect these points, what do you
find?
 The graph of an equation with only an x is a
horizontal line at that value.
Find the equation of the line with slope
of -2/3 and y-intercept of 5
 Y = mx + b
 Put the slope in for the m.
 Put the y-intercept in for the b
 Y = -2/3 x + 5
Find the equation of the line that
passes through (3, 5) and (-2, 4)
1.
Find the slope: (y2 – y1) / (x2 – x1)
(5 – 4)/(3 - -2) =
1/5
Use the slope and one point to find the b
y = mx + b
substitute point and
slope into equation
5 = (1/5)(3) + b solve for b
5 = 3/5 + b 22/5 = b
3. Write the equation
y = mx + b
y = 1/5 x + 22/5 put slope and y-intercept in
2.
Write the equation of the line passing
through (4,-2) and (-2, 4)
1.
Find slope:
1.
2.
2.
Find b: y = mx + b
1.
2.
3.
m = (y2 – y1) /( x2 – x1)
(4 - -2) / (-2 – 4) = 6/-6 = -1
Put slope and one of points in and solve for
b.
4 = (-1)(-2) + b
4=2+b
2=b
Write equation: y = mx + b
1.
2.
Put slope and y-intercept into equation.
y = -1x + 2
Horizontal Lines
 Find an equation of the horizontal line
containing the point (3,2)
 Horizontal lines – have no rise so y is
constant. Equations are in the form y = #
 In this example the equation would be y=2.
Vertical Lines
 Find the vertical line passing through (3,2)
 Vertical lines have no run so the x remains
constant. X = #
 In the example the equation would be x=3
Application
 Don receives $375 per week for selling new
and used cars at a car dealership in Oak
Lawn, Illinois. In addition he receives 5% of
the profit on any sales he generates. Write
an equation that relates his weekly salary, S,
when he has sales that generate a profit of x
dollars.
 S = 375 + .05x
 375 is set value with rate of change of .05 for
sales.
Assignment: Page 191
 #9, 13, 17, 21, 25, 31, 35, 39, 41, 47, 53, 67,
75, 79