Download Systems of Equations

Systems of Equations
Solve by Elimination
Solve by Elimination. . . .
System of equations: x + y = 12
-x + 3y = -8.
1. Both equations have to be in Standard
Form—the x’s and the y’s on one side of
the equation and the number by itself
on the other side.
2. The like terms have to be lined up.
System of equations: x + y = 12
-x + 3y = -8.

We can eliminate one of the variables—
x or y by adding the two equations
together.

Add the two equations together. What
do you get?
Solve by Elimination. . . .
x + y = 12
-x + 3y = -8
4y = 4
4 4
y= 1
y=1
What is x?
Substitute what we know for y.
y = 1 into one of our original
equations.
x + y = 12
x + 1 = 12
Solve for x.
x + 1 = 12
-1 -1
x = 11
y = 1 and x = 11
The solution to the system of equations:
x + y = 12
-x + 3y = -8
is (11, 1).
Now check your answer by substituting
into the original equations. . . .
(11, 1)
Remember it is (x, y).
x + y = 12
-x + 3y = -8
11 + 1 = 12
-11 + 3(1) = -8
Solve by Elimination. . . .
System of equations: 5x - 4y = -21
-2x + 4y = 18.
1. Both equations have to be in Standard
Form—the x’s and the y’s on one side of
the equation and the number by itself
on the other side.
2. The like terms have to be lined up.
System of equations: 5x - 4y = -21
-2x + 4y = 18.

We can eliminate one of the variables—
x or y by adding the two equations
together.

Add the two equations together. What
do you get?
Solve by Elimination. . . .
5x - 4y = -21
-2x + 4y = 18
3x = -3
3
3
x = -1
x = -1
What is y?
Substitute what we know for x,
x = -1, into one of our original
equations.
5x - 4y = -21
5(-1) + 4y = -21
Solve for y.
5(-1) - 4y = -21
-5 - 4y = -21
+5
+5
- 4y = -16
-4 -4
y=4
x = -1 and y = 4
The solution to the system of equations:
5x - 4y = -21
-2x + 4y = 18
is (-1, 4).
Now check your answer by substituting
into the original equations. . . .
(-1, 4)
Remember it is (x, y).
5x - 4y = -21
-2x + 4y = 18
5(-1) – 4(4) = -21
-2(-1) + 4(4) = 18
Solve by Elimination. . . .
System of equations: 2x + 7y = 31
5x - 7y = -45.
1.
2.
Both equations have to be in Standard
Form—the x’s and the y’s on one side of
the equation and the number by itself
on the other side.
The like terms have to be lined up.
System of equations: 2x + 7y = 31
5x - 7y = -45

We can eliminate one of the variables—
x or y by adding the two equations
together.

Add the two equations together. What
do you get?
Solve by Elimination. . . .
2x + 7y = 31
5x - 7y = -45
7x = -14
7
7
x = -2
x = -2
What is y?
Substitute what we know for x,
x = -2, into one of our original
equations.
2x + 7y = 31
2(-2) + 7y = 31
Solve for y.
2(-2) + 7y = 31
-4 + 7y = 31
+4
+4
7y = 35
7 7
y=5
x = -2 and y = 5
The solution to the system of equations:
2x + 7y = 31
5x - 7y = -45
is (-2, 5).
Now check your answer by substituting
into the original equations. . . .
(-2, 5)
Remember it is (x, y).
2x + 7y = 31
5x - 7y = -45
2(-2) + 7(5) = 31
5(-2) - 7(5) = -45
Solve by Elimination. . . .
System of equations: x + y = 30
x + 7y = 6.
1. Both equations have to be in Standard
Form—the x’s and the y’s on one side of
the equation and the number by itself
on the other side.
2. The like terms have to be lined up.
System of equations: x + y = 30
x + 7y = 6

We cannot simply add them together
because that does not eliminate one of the
variables.

We have to multiply one of the equations by
-1 and them add the two equations together.

Multiply one of the equations by -1 and then
add them together. What do you get?
Solve by Elimination. . . .
-1(x + y = 30)
-x - y = -30
x + 7y = 6
6y = -24
6 6
y = -4
y = -4
What is x?
Substitute what we know for y,
y = -4, into one of our original
equations.
x + y = 30
x + -4 = 30
Solve for x.
x + -4 = 30
+4 +4
x = 34
x = 34 and y = -4
The solution to the system of equations:
x + y = 30
x + 7y = 6
is (34, -4).
Now check your answer by substituting
into the original equations. . . .
(34, -4)
Remember it is (x, y).
x + y = 30
x + 7y = 6
34 + (-4) = 30
34 + 7(-4) = 6
Solve by Elimination. . . .
System of equations: x + y = 4
2x + 3y = 9.
1. Both equations have to be in Standard
Form—the x’s and the y’s on one side of
the equation and the number by itself
on the other side.
2. The like terms have to be lined up.
System of equations: x + y = 4
2x + 3y = 9

We cannot simply add them together
because that does not eliminate one of the
variables.

Multiply the first equation by -2 and then
add them together. What do you get?
Solve by Elimination. . . .
-2(x + y = 4)
-2x - 2y = -8
2x + 3y = 9
1y = 1
y= 1
y=1
What is x?
Substitute what we know for y,
y = 1, into one of our original
equations.
x+y=4
x+1=4
Solve for x.
x+1=4
-1 -1
x=3
x = 3 and y = 1
The solution to the system of equations:
x+y=4
2x + 3y = 9
is (3, 1).
Now check your answer by substituting
into the original equations. . . .
(3, 1)
Remember it is (x, y).
x+y=4
2x + 3y = 9
3+1= 4
2(3) + 3(1) = 9
System of equations: 2x + y = 20
6x - 5y = 12

We cannot simply add them together
because that does not eliminate one of the
variables.

Multiply the first equation by 5 and then add
them together. What do you get?
Solve by Elimination. . . .
5(2x + y = 20)
10x + 5y = 100
6x - 5y = 12
16x = 112
x= 7
x=7
What is y?
Substitute what we know for x,
x = 7, into one of our original
equations.
2x + y = 20
2(7) + y = 20
Solve for x.
14 + y = 20
-14
-14
y=6
x = 7 and y = 6
The solution to the system of equations:
2x + y = 20
6x - 5y = 12
is (7, 6).
Now check your answer by substituting
into the original equations. . . .
(7, 6)
Remember it is (x, y).
2x + y = 20
6x - 5y = 12
2(7) + 6 = 20
6(7) - 5(6) = 12
System of equations: -5x – 8y = 17
2x - 7y = -17

We cannot simply add them together
because that does not eliminate one of the
variables.

Multiply the first equation by 2 and the
second equation by 5 and then add them
together. What do you get?
Solve by Elimination. . . .
2(-5x - 8y = 17)
5(2x – 7y = -17)
-10x – 16y = 34
10x - 35y = -85
-51y = -51
y= 1
y=1
What is x?
Substitute what we know for x,
y = 1, into one of our original
equations.
-5x - 8y = 17
-5x – 8(1) = 17
Solve for x.
-5x – 8 = 17
+8 +8
-5x = 25
x = -5 and y = 1
The solution to the system of equations:
-5x - 8y = 17
2x - 7y = -17
is (-5, 1).
Now check your answer by substituting
into the original equations. . . .
(-5, 1)
Remember it is (x, y).
-5x - 8y = 17
2x - 7y = -17
-5(-5) – 8(1) = 17
2(-5) - 7(1) = -17
For Elimination






1. First the two equations need to be in
standard form and the like terms added up.
2. Then multiply one or both equations until
you get two opposite like terms.
3. Add the two equations.
4. Solve for the first variable.
5. Plug the first variable back in to find the
other variable.
6.Plug your answers back into the original
equation to solve.