There are several ways to solve a system of linear equations. This article focuses on 4 methods:

- Graphing
- Substitution
- Elimination: Addition
- Elimination: Subtraction

### 1. Solve a System of Equations by Graphing

Find the solution to the following system of equations:

y=x+ 3y= -1x- 3

*Note: Since the equations are in slope-intercept form, solving by graphing is the best method.*

**1. Graph both equations.**

**2. Where do the lines meet?** (-3, 0)

**3. Verify that your answer is correct. Plug x = -3 and y = 0 into the equations.**

y=x+ 3

(0) = (-3) + 3

0 = 0

Correct!

y= -1x- 3

0 = -1(-3) - 3

0 = 3 - 3

0 = 0

Correct!

### 2. Solve a System of Equations by Substitution

Find the intersection of the following equations. (In other words, solve for *x* and *y*.)

3

x+y= 6x= 18 -3y

*Note: Use the Substitution method because one of the variables, x, is isolated. *

**1. Since x is isolated in the top equation, replace x in the top equation with 18 - 3y. **

3 (

18 – 3) +yy= 6

**2. Simplify.**

54 – 9

y+y= 6

54 – 8y = 6

**3. Solve.**

54 – 8y– 54 = 6 – 54

-8y= -48

-8y/-8 = -48/-8= 6

y

**4. Plug in y = 6 and solve for x. **

x= 18 -3y= 18 -3(6)

x= 18 - 18

x= 0

x

**5. Verify that (0,6) is the solution.**

x= 18 -3y

0 = 18 – 3(6)

0 = 18 -18

0 = 0

### 3. Solve a System of Equations by Elimination (Addition)

Find the solution to the system of equations:

x+y= 180

3x+ 2y= 414

*Note: This method is useful when 2 variables are on one side of the equation, and the constant is on the other side.*

**1. Stack the equations to add.**

**2. Multiply the top equation by -3.**

-3(x + y = 180)

**3. Why multiply by -3? Add to see.**

-3x + -3y = -540

+ 3x + 2y = 414

0 + -1y = -126

*Notice that x is eliminated.*

**4. Solve for y:**

y= 126

**5. Plug in y = 126 to find x.**

x+y= 180

x+ 126 = 180

x= 54

**6. Verify that (54, 126) is the correct answer.**

3

x+ 2y= 4143(54) + 2(126) = 414

414 = 414

### 4. Solve a System of Equations by Elimination (Subtraction)

Find the solution to the system of equations:

y- 12x= 3y- 5x= -4

*Note: This method is useful when 2 variables are on one side of the equation, and the constant is on the other side.*

**1. Stack the equations to subtract.**

y- 12x= 3

0 - 7x= 7

*Notice that y is eliminated.*

**2. Solve for x.**

-7

x= 7x= -1

**3. Plug in x = -1 to solve for y.**

y- 12x= 3y- 12(-1) = 3y+ 12 = 3y= -9

**4. Verify that (-1, -9) is the correct solution.**

(-9) - 5(-1) = -4

-9 + 5 = -4