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Foundations of Mathematics
CaLc Math
Linear Equations in a Nutshell
fx = -0.5x-1
General Equation
for a line:
gx = 2x+1
8
hx = -0.5x-4
qx = -0.5x+4
y = mx + b
6
4
Slope:
2
Method 1: Counting the Squares
rise
slope  m 
run
-10
Example 
m=
-5
5
-2
NOTE:
-4
Method 2: The equation
-6
y2  y1
slope  m 
E.g. What’s the slope of a line that passes through (-2, -5) and (5, 2)?
x2  x1
-8
NOTE:
What if the equation isn’t in “y = mx + b” form? You should
Example 1: 2x + 4y = 8
Example 2: 8x – y = 10
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Tuesday, June 27, 2017
Linear Equations in a nutshell
page 1 of 5
8
Foundations of Mathematics
fx = -0.5x-1
CaLc Math
6
g x = 2x+1
8
Special Slope properties:
h x = -0.5x-4
4
q x = -0.5x+4
Parallel
Lines
6
Perpendicular
Lines
2
4
2
-5
5
-2
-10
10
-5
5
-2
-4
-4
-6
slope of line 3: m =
slope of line 1: m =
-6
-8
slope of line 2: m =
slope of line 4: m =
Conclusion:
Conclusion:
Example 1:
Write the equation of the line that is
perpendicular to y = ¾ x + 5 and has a y-int.
of (0,1)?
Example 2:
What is the equation of a line is parallel to a
line that passes through (2,8) and (4, 5) and
has a y-intercept of (0, -15)?
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Tuesday, June 27, 2017
-8
Linear Equations in a nutshell
page 2 of 5
1
Foundations of Mathematics
CaLc Math
Graphing: There are 2 ways to graph a line:
Method 1: Table of values. Example: Graph y = - 2x + 1
X
Y = -2x + 1
10
8
6
4
2
-10
-5
5
10
5
10
-2
-4
-6
10
-8
Method 2: y = mx + b method
8
Graph y = - ¾ x – 2
6
Slope:
4
2
y-intercept:
-10
-5
-2
-4
-6
-8
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Tuesday, June 27, 2017
Linear Equations in a nutshell
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Foundations of Mathematics
CaLc Math
Solutions to linear systems:
In the algebra unit, we used elimination and substitution to solve for the (x, y). Another method
is graphing, and where the two lines meet, it is the solution.
Examples:
Part 1
2
 y  x 2
3
1
 y x1
3
Solution: (x, y) = (
Part 2 ** REARRANGE EQ’ns to y=mx+b!
 x – 2y +8 = 0
 -x + 4y =24
,
)
Solution: (x, y) = (
,
)
10
8
6
4
2
-10
-5
5
10
-2
-4
-6
-8
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Tuesday, June 27, 2017
Linear Equations in a nutshell
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Foundations of Mathematics
CaLc Math
Practice Questions:
Question 1: Using the 2nd method of graphing lines.
identify the b and then use the slope!)
2
a. y   x  7
5
b. y  2 x
4
c. y  x  2
3
Graph these lines (Hint: Remember to first
2
x5
3
e. y=-4x -3
f. –1x + 4y =8 (Rearrange this equation
to get y =mx+b form)
d. y 
Question 2: Graph these systems of equations to find the SOLUTION (where they intersect)
a)  y = 2x – 4
b)  y = -1x + 3
 y = 3x – 6
y=x+1
c)  y = 2x + 1
 y = - ½x + 1
d)  y = 7x - 2
 y = -x + 6
e)  y = 2x + 1
 y = 3x + 5
f)  y = -x – 4
y=x
Question 3:
a. Write the equation of the line that is parallel to y = 3x + 6 and has a y-intercept of (0,-3).
b. A line is perpendicular to y = 7x – 12 and passes through (0, 3). What is the equation?
c. A line parallel to the line y = - ½ x + 9 and passes through (0, 19). What is the equation?
d. A line is perpendicular to y = - ½ x + 9 and passes through (0, 19). What is the equation?
e. There is a line is parallel to the line that passes through (9, 10) and (11, 12). It has a yintercept of (0, -20). What is the equation?
f. There is a line that is perpendicular to the line that passes through (-3, -5) and (1, 2); the
perpendicular line has b = 3. What is the equation of the line?
Question 4: Rearrange the equation to create the y = mx + b form. Then identify your m & b
a. y + 7x = 12
e. 4x – y = 9
b. x + 3y = 7
f. 8y – x = -4
c. y – 8 = 5x
g. 3x + y – 21
d. x – 7y = 10
h. x – 7y = 0
Question 5: Graph these equations to find the SOLUTION of the linear system. (Rearrange
these equations to the y=mx+b form before you graph them.)
a.  y = 3x - 5
b.  3x – y = 4
 2x – 4y = 10
y=x+1
c.  -2y = x – 8
 2x – 4y = - 16
d.  y = -2x - 6
 -x – 3y = 13
e.  4x – 5 = y
 x – 4y = 5
f.  y = -4x – 3
 2x – y = 3
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Tuesday, June 27, 2017
Linear Equations in a nutshell
page 5 of 5