2. Warm-Up
1. A 66 ft. board is cut into three pieces. The second piece is
1.5 times longer than the first. The third piece is twice as
long as the second. How long is each piece?
4. Solve and graph the following: -5< x - 4 <2
5. Solve and graph the following: -3h < 19 or 7h - 3> 18
Class Notes & Practice Problems Section of Notebook
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3.
4. Graphs of y = b
Where b is any number
y = b are always horizontal lines,
and are functions
Graphs of x = a
Where a is any number
x = a are always vertical lines, and
are not functions
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5. Graphs of y = mx
y = "a number times x"
y = mx lines will always go through
the origin, and will be at the angle
shown.
Graphs of y = mx + c
y = "a number times x plus c"
y = mx + c are similar in shape to y = mx,
but do not go through the origin.
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6. Defining Linear Equations
HOW DO I KNOW IF AN EQUATION IS
LINEAR?
If an equation is linear, a constant
change in the x-value corresponds to a
constant change in the y-value.
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7. Solutions to Linear Equations
To ask "Is the ordered pair (1,3) a solution to the equation
y = 10x - 7?", is the same as asking, " Is the point (1,3) on
the line of y = 10x -7"? How do we know?
Determine whether the following ordered pairs are solutions to
the equation y = 10x -7
a.) (1,3) b.) (2, 13) c.) (-1,-3)
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9. The slope of the line passing through the two
points (x1, y1) and (x2, y2) is given by the formula
y
(x2, y2)
y2 – y1
change in y
(x1, y1) x2 – x1
change in x
x
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10. Example: Find the slope of the line passing
through the points (2, 3) and (4, 5).
Use the slope formula with x1= 2, y1 = 3, x2 = 4, and y2 = 5.
y2 – y1 5–3 2
m= = = =1
x2 – x1 4–2 2
y
(4, 5)
(2, 3) 2
2
x
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11. Finding the x-and y-intercepts of Linear Equations
What does it mean to INTERCEPT a pass in football?
The path of the defender crosses the path of the thrown
football.
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12. The x-intercept is where
(2, 0)
the graph crosses the x-
axis.
The y-coordinate is
always zero.
The y-intercept is where
the graph crosses the y-
axis. (0, 6)
The x-coordinate is
always zero.
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23. A linear equation written in the form y = mx + b is in
slope-intercept form.
The slope is m and the y-intercept is (0, b).
To graph an equation in slope-intercept form:
1. Write the equation in the form y = mx + b. Identify m and b.
2. Plot the y-intercept (0, b).
3. Starting at the y-intercept, find another point on the line
using the slope.
4. Draw the line through (0, b) and the point located using the
slope.
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24. Example: Graph the line y = 2x – 4.
1. The equation y = 2x – 4 is in the slope-intercept form.
So, m = 2 and b = - 4. y
2. Plot the y-intercept, (0, -
4). x
change in y 2
3. The slope is 2. m = =
change in x 1 (1, -2)
4. Start at the point (0, 4). 2
(0, - 4)
Count 1 unit to the right and 2 units up 1
to locate a second point on the line.
The point (1, -2) is also on the line.
5. Draw the line through (0, 4) and (1, -2).
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25. A linear equation written in the form y – y1 = m(x – x1)
is in point-slope form.
The graph of this equation is a line with slope m
passing through the point (x1, y1).
Example: y
The graph of the equation 8 m=-
1
2
y – 3 = - 1 (x – 4) is a line
2 4 (4, 3)
of slope m = - 1 passing
2
through the point (4, 3). x
4 8
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26. Example: Write the slope-intercept form for the equation
of the line through the point (-2, 5) with a slope of 3.
Use the point-slope form, y – y1 = m(x – x1), with m = 3 and
(x1, y1) = (-2, 5).
y – y1 = m(x – x1) Point-slope form
y – y1 = 3(x – x1) Let m = 3.
y – 5 = 3(x – (-2)) Let (x1, y1) = (-2, 5).
y – 5 = 3(x + 2) Simplify.
y = 3x + 11 Slope-intercept form
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27. Example: Write the slope-intercept form for the
equation of the line through the points (4, 3) and (-2,
5).
5–3 =- 2 =- 1 Calculate the slope.
m=
-2 – 4 6 3
y – y1 = m(x – x1) Point-slope form
1 1
y–3=- (x – 4) Use m = - and the point (4, 3).
3 3
y = - 1 x + 13 Slope-intercept form
3 3
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28. Two lines are parallel if they have the same slope.
If the lines have slopes m1 and m2, then the lines
are parallel whenever m1 = m2. y
(0, 4)
Example:
The lines y = 2x – 3
y = 2x + 4
and y = 2x + 4 have slopes
m1 = 2 and m2 = 2. x
y = 2x – 3
The lines are parallel.
(0, -3)
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29. Two lines are perpendicular if their slopes are
negative reciprocals of each other.
If two lines have slopes m1 and m2, then the lines
are perpendicular whenever
y
1
m2= - or m1m2 = -1. y = 3x – 1
m1
(0, 4) 1
Example: y=- x+4
3
The lines y = 3x – 1 and
1
y = - x + 4 have slopes
3 x
m1 = 3 and m2 = -1 .
3 (0, -1)
The lines are perpendicular.
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30. Equations of the form ax + by = c are called
linear equations in two variables.
y
This is the graph of the
equation 2x + 3y = 12.
x
-2 2
The point (0,4) is the y-intercept.
The point (6,0) is the x-intercept.
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