Math Skill: System of Equations Addition
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A system of equations is a set of equations that are linked together. The simplest system of equations is two equations with two unknowns.

One way to solve a system of equations is by addition:

1. Prepare the equations so that there is one variable in both equations that has the same coefficient.
This can be done by multiplying one or both equations by a constant.
2. Use addition or subtraction to eliminate one of the variables.
3. Solve for the other variable.
4. Plug the value into one of the original equations to solve for the first variable.

Why do steps 1 & 2 work? Remember that an equation means are the two sides are equal. And, if you do the same thing to both sides, they should still be equal.

Solve. Answer in the form (x,y). For example: (-2,3)
 x - y = 6-x + 2y = - 9 Answer (x,y):

Step 1: Prepare the equations
When you add x from the first equation and -x from the second equation, the variable x is eliminated.
So nothing needs to be done to prepare the equations.

Steps 2 & 3: Add the two equations and solve for y:

 x - y = 6 -x + 2y = - 9 y = - 3

Step 4: Now solve for x.

 x - y = 6 x - (-3) = 6 x + 3 = 6 x + 3 - 3 = 6 - 3 x = 3

Example: 2 Equation preparation with subtraction

Solve. Answer in the form (x,y). For example: (-2,3)
 x - 2y = 214x - 5y = 54 Answer (x,y):

Step 1: Prepare the equations
Since the coefficients do not match for either variable in the two equations, we will need to use multiplication.
Let's multiply the first equation by 4 to get

4x - 8y = 84
4x - 5y = 54

Steps 2 & 3: Subtract and solve for y

 4x - 8y = 84 - (4x - 5y = 54) 4x - 8y = 84 -4x + 5y = -54 -3y = 30 y = -10

Step 4: Now solve for x.

 x - 2y = 21 x - 2(-10) = 21 x + 20 = 21 x + 20 - 20 = 21 - 20 x = 1

### Complexity=3

Solve. Answer in the form (x,y). For example: (-2,3)
 1.   x - 3y = 63x + y = - 2 Answer (x,y): 2.   x + y = 2-x + y = 0 Answer (x,y):

### Complexity=5

Solve. Answer in the form (x,y). For example: (-2,3)
 1.   x - y = 3x + 5y = - 3 Answer (x,y): 2.   5x - 2y = 7- 4x - 5y = 1 Answer (x,y):

### Complexity=10

Solve. Answer in the form (x,y). For example: (-2,3)
 1.   - 8x + 5y = - 3- 4x - 9y = 79 Answer (x,y): 2.   - 7x - 8y = 1- 2x + 5y = - 7 Answer (x,y):

### Complexity=13

Solve. Answer in the form (x,y). For example: (-2,3)
 1.   5x - 6y = 59x + 13y = 247 Answer (x,y): 2.   - 2x - y = 2x + 6y = 65 Answer (x,y):

### Complexity=14

Solve. Answer in the form (x,y). For example: (-2,3)
 1.   - 11x + y = 76- 7x + 6y = 43 Answer (x,y): 2.   - 8x - 5y = 66- 12x - y = 34 Answer (x,y):

### Complexity=15

Solve. Answer in the form (x,y). For example: (-2,3)
 1.   - 11x - 15y = 3383x + y = - 52 Answer (x,y): 2.   3x - 8y = - 114x + 5y = 95 Answer (x,y):

### Complexity=3

Solve. Answer in the form (x,y). For example: (-2,3)
1
x - 3y = 6
3x + y = - 2

Solution
x - 3y = 6
3x + y = - 2

Multiply the second equation by 3
x - 3y = 6
9x + 3y = - 6

Add the equations to eliminate y.
x - 3y = 6
+ [ 9x + 3y = - 6 ]
10x = 0

Now solve for x
Divide by 10

x = 0

Now plug value of x into the original first equation
0 - 3y = 6
- 3y = 6
Divide by - 3

y = - 2

2
x + y = 2
-x + y = 0

Solution
x + y = 2
-x + y = 0

Add the equations to eliminate x.
x + y = 2
+ [ -x + y = 0 ]
2y = 2

Now solve for y
Divide by 2

y = 1

Now plug value of y into the original first equation
x + 1 = 2
x + 1 = 2
x + 1 - 1 = 2 - 1
x = 1

### Complexity=5

Solve. Answer in the form (x,y). For example: (-2,3)
1
x - y = 3
x + 5y = - 3

Solution
x - y = 3
x + 5y = - 3

Subtract the equations to eliminate x.
x - y = 3
- [ x + 5y = - 3 ]
- 6y = 6

Now solve for y
Divide by - 6

y = - 1

Now plug value of y into the original first equation
x - - 1 = 3
x + 1 = 3
x + 1 - 1 = 3 - 1
x = 2

2
5x - 2y = 7
- 4x - 5y = 1

Solution
5x - 2y = 7
- 4x - 5y = 1

Multiply the first equation by 5
Multiply the second equation by 2
25x - 10y = 35
- 8x - 10y = 2

Subtract the equations to eliminate y.
25x - 10y = 35
- [ - 8x - 10y = 2 ]
33x = 33

Now solve for x
Divide by 33

x = 1

Now plug value of x into the original first equation
5 × 1 - 2y = 7
- 2y + 5 = 7
- 2y + 5 - 5 = 7 - 5
- 2y = 2

Divide by - 2

y = - 1

### Complexity=10

Solve. Answer in the form (x,y). For example: (-2,3)
1
- 8x + 5y = - 3
- 4x - 9y = 79

Solution
- 8x + 5y = - 3
- 4x - 9y = 79

Multiply the second equation by 2
- 8x + 5y = - 3
- 8x - 18y = 158

Subtract the equations to eliminate x.
- 8x + 5y = - 3
- [ - 8x - 18y = 158 ]
23y = - 161

Now solve for y
Divide by 23

y = - 7

Now plug value of y into the original first equation
- 8x + 5 × - 7 = - 3
- 8x - 35 = - 3
- 8x - 35 + 35 = - 3 + 35
- 8x = 32

Divide by - 8

x = - 4

2
- 7x - 8y = 1
- 2x + 5y = - 7

Solution
- 7x - 8y = 1
- 2x + 5y = - 7

Multiply the first equation by 2
Multiply the second equation by 7
- 14x - 16y = 2
- 14x + 35y = - 49

Subtract the equations to eliminate x.
- 14x - 16y = 2
- [ - 14x + 35y = - 49 ]
- 51y = 51

Now solve for y
Divide by - 51

y = - 1

Now plug value of y into the original first equation
- 7x - 8 × - 1 = 1
- 7x + 8 = 1
- 7x + 8 - 8 = 1 - 8
- 7x = - 7

Divide by - 7

x = 1

### Complexity=13

Solve. Answer in the form (x,y). For example: (-2,3)
1
5x - 6y = 5
9x + 13y = 247

Solution
5x - 6y = 5
9x + 13y = 247

Multiply the first equation by 9
Multiply the second equation by 5
45x - 54y = 45
45x + 65y = 1235

Subtract the equations to eliminate x.
45x - 54y = 45
- [ 45x + 65y = 1235 ]
- 119y = - 1190

Now solve for y
Divide by - 119

y = 10

Now plug value of y into the original first equation
5x - 6 × 10 = 5
5x - 60 = 5
5x - 60 + 60 = 5 + 60
5x = 65

Divide by 5

x = 13

2
- 2x - y = 2
x + 6y = 65

Solution
- 2x - y = 2
x + 6y = 65

Multiply the second equation by 2
- 2x - y = 2
2x + 12y = 130

Add the equations to eliminate x.
- 2x - y = 2
+ [ 2x + 12y = 130 ]
11y = 132

Now solve for y
Divide by 11

y = 12

Now plug value of y into the original first equation
- 2x - 12 = 2
- 2x - 12 = 2
- 2x - 12 + 12 = 2 + 12
- 2x = 14

Divide by - 2

x = - 7

### Complexity=14

Solve. Answer in the form (x,y). For example: (-2,3)
1
- 11x + y = 76
- 7x + 6y = 43

Solution
- 11x + y = 76
- 7x + 6y = 43

Multiply the first equation by 6
- 66x + 6y = 456
- 7x + 6y = 43

Subtract the equations to eliminate y.
- 66x + 6y = 456
- [ - 7x + 6y = 43 ]
- 59x = 413

Now solve for x
Divide by - 59

x = - 7

Now plug value of x into the original first equation
- 11 × - 7 + y = 76
y + 77 = 76
y + 77 - 77 = 76 - 77
y = - 1

2
- 8x - 5y = 66
- 12x - y = 34

Solution
- 8x - 5y = 66
- 12x - y = 34

Multiply the second equation by 5
- 8x - 5y = 66
- 60x - 5y = 170

Subtract the equations to eliminate y.
- 8x - 5y = 66
- [ - 60x - 5y = 170 ]
52x = - 104

Now solve for x
Divide by 52

x = - 2

Now plug value of x into the original first equation
- 8 × - 2 - 5y = 66
- 5y + 16 = 66
- 5y + 16 - 16 = 66 - 16
- 5y = 50

Divide by - 5

y = - 10

### Complexity=15

Solve. Answer in the form (x,y). For example: (-2,3)
1
- 11x - 15y = 338
3x + y = - 52

Solution
- 11x - 15y = 338
3x + y = - 52

Multiply the second equation by 15
- 11x - 15y = 338
45x + 15y = - 780

Add the equations to eliminate y.
- 11x - 15y = 338
+ [ 45x + 15y = - 780 ]
34x = - 442

Now solve for x
Divide by 34

x = - 13

Now plug value of x into the original first equation
- 11 × - 13 - 15y = 338
- 15y + 143 = 338
- 15y + 143 - 143 = 338 - 143
- 15y = 195

Divide by - 15

y = - 13

2
3x - 8y = - 11
4x + 5y = 95

Solution
3x - 8y = - 11
4x + 5y = 95

Multiply the first equation by 4
Multiply the second equation by 3
12x - 32y = - 44
12x + 15y = 285

Subtract the equations to eliminate x.
12x - 32y = - 44
- [ 12x + 15y = 285 ]
- 47y = - 329

Now solve for y
Divide by - 47

y = 7

Now plug value of y into the original first equation
3x - 8 × 7 = - 11
3x - 56 = - 11
3x - 56 + 56 = - 11 + 56
3x = 45

Divide by 3

x = 15