1. Today:

2. Grades & Weighting:
All Quarters
# of Assignments
Weight
Class Work
10 Graded
20%
Home Work
Tests
Notebook
6
6
1
20%
40%
10%
Final Exam
1
10%

3. 5 Most Missed Questions from Final Exam
#5; Version 1: 39% correct
Mrs. Wright rents a car to see her friend Mrs. Wong. The cost
is $32.00 plus $.75 a mile. The trip is 45 miles each way. What
is the total cost for Wright to see Wong?
A. $32.75
B. $99.50
C. $75.00
D. $65.75
E. None
#4; Version 1: 37% correct
Solve for p: A = (ap + pH)
A) p = a + H
A
B) p =
A
C) p = A – a D) p = H - a
a+H
H
A
E. None

4. 5 Most Missed Questions from Final Exam
#3; Version 2: 36% correct
A. 125%
B. 90%
C. 55%
D. 80%
E. None

5. 5 Most Missed Questions from Final Exam
#2; Version 3: 29% correct
Bob made a fresh pot of coffee in the morning. By 10:00 am only 3
cups remained. If 90% of the coffee had been consumed, how
many cups of coffee did Bob make in total?
A. 22
B. 15
#1; Version 3: 26% correct
C. 18
D. 30
E. 20

9. SYSTEMS OF LINEAR EQUATIONS
So far, we have solved equations with one variable: 3x + 5 =
35
and in two variables. 3x + 5y =
That will change now as
35
In both cases, though
we solve multiple
we have only been able
equations at the same
to solve one equation at
time, looking for an
a time.
ordered pair which solves
each equation, and thus is
a solution for both.
Example:
There are 3 methods for solving systems of
3x + 3y = -3
equations: 1) by Graphing 2) By Elimination
y=x+1
3) By Substitution
Using any of the 3 systems will show the solution to these
equations is x = -1, y = 0

10. WHAT IS A SYSTEM OF LINEAR EQUATIONS?
If the system of linear equations is going to have a solution,
then the solution will be an ordered pair (x , y) where x and y
make both equations true at the same time.
If the lines are parallel, there will be no solutions. If the
equations are the same line, there will be an infinite
number of solutions. All other lines will have a single
solution.
There are several methods of solving systems of
equations; we'll look at a couple today.

11. Tell whether the ordered pair is a solution of
the given system.
(5, 2);
3x – y = 13
Substitute 5 for x and 2
for y in each equation in
the system.
3x – y 13
0
3(5) – 2 13
2–2 0
0 0
15 – 2 13
13 13
The ordered pair (5, 2) makes both equations true.
(5, 2) is the solution of the system.

12. Helpful Hint
If an ordered pair does not satisfy the first equation in the
system, there is no reason to check the other equations.
x + 3y = 4
(–2, 2); –x + y = 2
x + 3y = 4
–2 + 3(2) 4
–2 + 6 4
4 4
Substitute –2 for x and 2
for y in each equation in
the system.
–x + y = 2
– (–2) + 2 2
4 2
The ordered pair (–2, 2) makes one
equation true but not the other.
(–2, 2) is not a solution
of the system.

13. SOLVING LINEAR SYSTEMS BY GRAPHING
Consider the following system:
Using the graph to the right,
we can see that any of these
ordered pairs will make the
first equation true since they lie
on the line.
Notice that any of these points will
make the second equation true.
x – y = –1
x + 2y = 5
y
(1 , 2)
However, there is ONE
point that makes both true
at the same time…
The point where they intersect makes both equations
true at the same time.
x

14. Practice 1
Graph the system of equations. Determine whether the
system has one solution, no solution, or infinitely many
solutions. If the system has one solution, determine the
solution.

15. Practice 1
y
The two equations in slopeintercept form are:
x
Plot points for each line.
Draw the lines.
These two equations represent the same line.
Therefore, this system of equations has infinitely many solutions .

16. Practice 2
y
The two equations in
slope-intercept form are:
x
Plot points for each line.
Draw in the lines.
This system of equations represents two parallel lines.
This system of equations has no solution because these
two lines have no points in common.

17. Practice 3
The two equations in
slope-intercept form are:
y
x
Plot points for each line.
Draw in the lines.
This system of equations represents two intersecting lines.
The solution to this system of equations is a single point (3,0)

18. Graphing to Solve a Linear System
Let's summarize! There are 4 steps to solving a linear
system using a graph.
Step 1: Put both equations in
slope - intercept form.
Solve both equations for y, so that
each equation looks like
y = mx + b.
Step 2: Graph both equations on
the same coordinate plane.
Use the slope and y - intercept
for each equation in step 1.
Step 3: Plot the point where the
graphs intersect.
This is the solution! LABEL the
solution!
Step 4: Check to make sure your
solution makes both equations true.
Substitute the x and y values into
both equations to verify the point is
a solution to both equations.

19. Solve: by ELIMINATION
5x - 4y = -21
-2x + 4y = 18
3x + 0 = -3
x = -1
Like variables must be lined under each other.
We need to eliminate (get rid of) a variable by cancelling out
one of the variables. We then solve for the other variable.
The y’s be will the easiest. So, we will add the two equations.
THEN----

20. 5x - 4y = -21
5(-1) – 4y = -21
Substitute your first
solution into either original -5 – 4y = -21
equation and solve for the
5
5
second variable.
-4y = -16 y = 4
The solution to this system
of equations is:
(-1, 4)
Now check your answers
in both equations------

21. 5X - 4Y = -21
5(-1) – 4(4) = -21
-5 - 16 = -21
-21 = -21
-2x + 4y = 18
-2(-1) + 4(4) = 18
2 + 16 = 18
18 = 18

22. Solve:
by ELIMINATION
x + y = 30
x + 7y = 6
We need to eliminate (get rid of) a variable.
To simply add this time will not eliminate a variable. If
one of the x’s was negative, it would be eliminated when
we add. So we will multiply one equation by a – 1.

23. X + Y = 30
( X + 7Y = 6) -1
Now add the two
equations and solve.
-X – 7Y = - 6
-6Y = 24
-6
-6
Y=-4
THEN----

24. Substitute
your answer
into either
original
equation and
solve for the
second
variable.
X + Y = 30
X + - 4 =30
4 4
X = 34
Solution
(34, - 4)
Now check our answers
in both equations------

25. x + y = 30
34 + - 4 = 30
30 = 30
x + 7y = 6
34 + 7(- 4) = 6
34 - 28 = 6
6=6

26. Solve: Elimination By
Multiplying
Like variables
must be lined
under each other.
x + +y0y 4 4
0x = =
2x + 3y = 9
We need to eliminate (get rid of) a variable.
To simply add this time will not eliminate a variable. If there was a –
2x in the 1st equation, the x’s would be eliminated when we add. So
we will multiply the 1st equation by a – 2.

27. (X + Y = 4 ) -2
-2X - 2 Y = - 8
2X + 3Y = 9
2X + 3Y = 9
Now add the two
equations and solve.
THEN----
Y=1

28. Substitute your answer into either original
equation and solve for the second variable.
X+Y=4
X +1=4
- 1 -1
X=3
Solution
(3,1)
Now check our answers in both equations--

29. x+y=4
3+1=4
4=4
2x + 3y = 9
2(3) + 3(1) = 9
6+3=9
9=9

30. Class Work

31. Many states fine speeders $15 for each mile per hour over
the speed limit of 45 mph. Graph this relationship
a) Determine the dependent & independent variables
b) Write the equation representing fines as a relation to speed.
Test your equation with real numbers to see if it fits. (Your first
equation may not be the correct one, so try others.
c) Label the axes, plot the x-intercept and at least one other point.
d) What is the speed of the driver given a $180 ticket for
driving over the 45 MPH speed limit?
e) What is the fine for a driver going 68 MPH?
f) Is this a function?
g) Is this a linear equation?

32. Fine (cost)
e) What is the fine
for a driver going
68 MPH? $345
y = 15(x – 45)
250
f) Is this a
function? No
g) Is this a linear
equation? No
(60,225)
150
50
20
(45,0)
Speed

33. Odds n’ Ends

34. REVIEW: SOLVING BY GRAPHING
If the lines cross once, there will be one solution.
If the lines are parallel, there will be no solutions.
If the lines are the same, there will be an infinite
number of solutions.

36. Writing Equations of Lines
C. Given a point and the equation of a line parallel to it.
Find the equation of the line that passes throug
is parallel to 4x – 2y =3.
Rewrite the equation to slopeintercept form to get the slope.
Solution:
y1 = -5x1 = 1

37. Writing Equations of Lines
D. Given a point and equation of a line perpendicular to
it.
Find the equation of the line that passes through (1, -5) and is
perpendicular to 4x – 2y =3.
Solution:
x1 = 1
y1 = -5
Rewrite the equation to slope- intercept
form to get the slope.
m= -

39. Solve:
by ELIMINATION
2x + 7y = 31
5x - 7y = - 45
x = -2
7x + 0 = -14
Like variables must be lined under each other.
THEN----

40. Substitute
your answer
into either
original
equation and
solve for the
second
variable.
2X + 7Y = 31
2(-2) + 7y = 31
-4 + 7y = 31
4
4
7y = 35
y=5
(-2, 5)
Now check our answers
in both equations-----Answer

41. 2x + 7y = 31
2(-2) + 7(5) = 31
-4 + 35 = 31
31 = 31
5x – 7y = - 45
5(-2) - 7(5) = - 45
-10 - 35 = - 45
- 45 =- 45

42. Solve: By Substitution
Recall that when we 'solve' a point-slope formula, we
end up in slope-intercept form. In much the same
way, the substitution method is closely related to the
elimination method.
After eliminating one variable and solving for the other,
we substitute the value of the variable back into the
equation. For example: Solve 2x + 3y = -26 using
elimination
What is the
value of x ?
4x - 3y =
2
At this point we substitute -4 for x,
and solve for y. This is exactly
what the substitution method is
except it is done at the beginning.

43. Solve: By Substitution
Example 1: y = 2x
4x - y = -4
Example 1: Substitute 2x for y in the 2nd equation
y = 2x
4x - 2x = -4; 2x = -4; x = -2
Then, substitute -2 for x in the first equation:
y = 2(-2); y = -4
Finally, plug both values in and check for equality.
-4 = 2(-2); True; 4(-2) - (-4) = -4; -8 + 4 = -4; True

44. Solve: By Substitution
Example 2: 3x + 5y = -7
x = 2y + 5
Example 3: y = 2x - 1
6x - 3y = 7

45. Applying Systems of Equations
Solve by Elimination.
Example 1: The sum of two numbers is 52. The
larger number is 2 more than 4 times the
smaller number. Find both numbers.
Example 1:
-(x ++y ==-52
-x - y 52)
x
52
x = 4y=+2
-4y
+ _________2 Rearrange
-5y = -50 y = 10
x + 10 = 52; x = 42

46. 3x - 2y = 8
2x - 2y = 4
3x - 2y = 8
-6x - 5y = -12
3x - 2y = 8
4x + y = 2

47. Class Work: