This document appears to be notes from a math class covering systems of inequalities and equations. It includes examples of solving systems of equations by elimination and substitution. It provides step-by-step instructions for setting up and solving systems using both methods. It also shows examples of applying systems of equations to word problems involving tickets sales and math test points.

2. February
2014
6
7
8
Review
Systems of Systems of
Inequalities Inequalities Coordinate
Unit
Help
With
Laundry
Rake
Leaves
11
13 Begin
14
15
Review for 12 Test:
Systems Monomial Monomials Exponents
Systems
Exponents
Eq./Ineq.
Unit
Test
No Warm-Up;
Practice Problem Section of Notebook

3. Solve Systems of Equations by Elimination
(Multiplying)
Like variables
must be lined
under each
other.
x + +y1y 4 4
1x = =
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.

4. Solve Systems of Equations by Elimination
(Multiplying)
( X + Y = 4) -2
2X + 3Y = 9
-2X - 2 Y = - 8
2X + 3Y = 9
Now add the two
equations and solve.
THEN----
Y=1

5. Solve Systems of Equations by Elimination
(Multiplying)
X+Y=4
X +1=4
- 1 -1
X=3
Solution
Substitute your
answer into either
original equation
and solve for the
second variable.
(3,1)
Now check our answers in both equations--

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

7. Solve Systems of Equations by Elimination
(Multiplying)
3x – 2y = -7
2x -5y = 10
Can you multiply either equation
by an integer in order to eliminate
one of the variables?
Here, we must multiply both
equations by a (different)
number in order to easily
eliminate one of the variables.
 Eliminate
 Plug back in
 Solve for other
variable
Multiply the top
equation by 2, and the
bottom equation by -3
Write your solution as
an ordered pair
(-5,-4)
Plug both solutions into
original equations

8. 3x – 2y = -7
-15 – (-8) = -7
-7 = - 7
2x - 5y = 10
-10 – (-20) = 10
10= 10

9. 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
4x - 3y = 2
What is the
value of x ?
-4
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.

10. Solve: By Substitution
Example 1: y = 2x
4x - y = -4
Since the first equation tells us
that y = 2x, replace the y with 2x
in the second equation.
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

11. Solve: By Elimination
x – 2y = 7
3x + 2y = 13
no
solution
Solve: By Substitution
3x + 2y = 8
2x – 3y = – 38
infinitely many solutions

12. Applying systems of Equations:
A math test has a total of 25 problems. Some
problems are worth 2 points and some are worth 3
points. The whole test is worth 63 points. How
many 2-point problems were there?
1. Mark
the text.
x=
# of 2-point
problems
2. Label
variables.
y=
3. Create
equations.
# of 3-point
problems
4. Solve.
12 2-point problems
5. Check.

13. Applying Systems of Equations:
Prom tickets cost $10 for singles and $15 for couples.
Fifty more couples tickets were sold than were singles
tickets. Total ticket sales were $4000.
How many of each ticket type were sold?

14. Applying Systems of Equations
After reading, determine the best method to solve
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

15. Class Work: