Elimination Using Addition and Subtraction
1. The sum of two numbers is 31. The greater number is 5 more
than the lesser number. What are the two numbers?
Let x = the greater number
Let y = the lesser number
x  y  31
x  y 5
y  5  y  31
2 y  26
y  13
x  18
2. Complementary angles have a sum of 90°. Angles A and B are
complementary and the measure of angle A is 14 ° less than
the measure of angle B. Find the measures of angles A and B.
y  52
x  y  90
y  14  y  90
Let x = the measure of A

2
y

104
Let
y
=
the
measure
of
B
x

y

14
x

38
3. What value of c in the system of equations 5x + y = – 1 and
– 2y = 10x + c will produce a system of equations with
infinitely many solutions?
5x  y  1
y  5x  1
 2 y  10x  2
c2
Elimination Using Addition and Subtraction
Elimination Using Addition and Subtraction
Sometimes adding two equations together
will eliminate one variable.
Using this step to solve a system of
equations is called elimination.
Elimination Using Addition
Use elimination to solve the system of equations.
Since the coefficients of the x terms, –3 and 3, are
additive inverses, you can eliminate the x terms by
adding the equations.
Write the equation in column
form and add.
Notice that the x value is eliminated.
Divide each side by –2.
Simplify.
Elimination Using Addition
Now substitute –15 for y in either equation to find the
value of x.
First equation
Replace y with –15.
Simplify.
Add 60 to each side.
Simplify.
Divide each side by –3.
Simplify.
Answer: The solution is (–24, –15).
Elimination Using Addition
Use elimination to solve the system of equations.
Answer: (2, 1)
Write and Solve a System of Equations
Four times one number minus three times another
number is 12. Two times the first number added to
three times the second number is 6. Find the numbers.
Let x represent the first number and y represent the
second number.
Four times
one number
4x
Two times
the first number
2x
minus
three times
another number
is
12.
–
3y
=
12
added to
three times
the second number
is
6.
+
3y
=
6
Write and Solve a System of Equations
Use elimination to solve the system.
Write the equation in column
form and add.
Notice that the y value is eliminated.
Divide each side by 6.
Simplify.
Write and Solve a System of Equations
Now substitute 3 for x in either equation to find the
value of y.
First equation
Replace x with 3.
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by –3.
Simplify.
Answer: The numbers are 3 and 0.
Write and Solve a System of Equations
Four times one number added to another number is
–10. Three times the first number minus the second
number is –11. Find the numbers.
Answer: –3, 2
Elimination Using Addition and Subtraction
Sometimes subtracting one equation from
another will eliminate one variable.
Elimination Using Subtraction
Use elimination to solve the system of equations.
–
Since the coefficients of the x terms, 4 and 4, are the same,
you can eliminate the x terms by subtracting the equations.
Write the equation in column
form and subtract.
Notice that the x value is eliminated.
Divide each side by 5.
Simplify.
Elimination Using Subtraction
Now substitute 2 for y in either equation to find the
value of x.
Second equation
Simplify.
Add 6 to each side.
Simplify.
Divide each side by 4.
Simplify.
Answer: The solution is (6, 2).
Elimination Using Subtraction
Use elimination to solve the system of equations.
Answer: The solution is (2, –6).
Elimination Using Subtraction
Multiple-Choice Test Item
If
A (3, –8)
and
B3
what is the value of y?
C –8
D (–8, 3)
Read the Test Item
You are given a system of equations, and you are
asked to find the value of y.
Elimination Using Subtraction
Solve the Test Item
You can eliminate the y terms by subtracting one equation
from the other.
Write the equation in column
form and subtract.
Notice that the y value is eliminated.
Divide each side by 14.
Simplify.
Elimination Using Subtraction
Now substitute 3 for x in either equation to solve for y.
First equation
Simplify.
Subtract 24 from each side.
Simplify.
Notice that B is the value of x and A is the solution of the
system of equations. However, the question asks for the
value of y.
Answer: C
Elimination Using Subtraction
Multiple-Choice Test Item
If
A4
Answer: D
and
B (4, –4)
what is the value of x?
C (–4, 4)
D –4