## Solving System of Equations by Substitution

#### Introduction:

#### Substitution:

In algebra, “Substitution” means putting numbers where the letters are.

#### Solve by Substitution:

The **substitution method** for solving a system of equations can be used to find the solution to a system. The goal is to obtain one equation containing only one variable.

#### Steps to Solve a Linear System of Equations by Substitution:

- Solve for a variable in either equation. If possible, solve for a variable with a numerical coefficient of 1 to avoid working with fractions.
- Substitute the expression found for the variable in Step 1 into the other equation. This will result in an equation containing only one variable.
- Solve the equation obtained in step 2.
- Substitute the value found in step 3 into the equation from step 1. Solve the equation to find the remaining variable.
- Check your solution in all equations in the system.

### Solve System of Equations Using Substitution

**Example 1:**

What is the solution to the system of equations?

x + y = 8

x + 2y = 10

**Solution:**

x + y = 8 ……….. (Equation One)

x + 2y = 10……… (Equation Two)

**Step 1:** Select one equation. Solve for one variable to be isolated on one side of the equal sign.

x + y = 8

Subtract y from both sides

x + y – y = 8 – y

x = 8 – y

**Step 2:** Substitute this expression into the second equation to find the value of the one variable.

Equation one: x = 8 – y

Equation two: x + 2y = 10

New equation: (8 – y) + 2y = 10

Combine like Terms: (8 – y) + 2y = 10

8 + y = 10

Subtract 8 from both sides

8 + y – 8= 10 – 8

y = 2

**Step 3:** Substitute this value to find the value of the other variable.

y = 2

x + y = 8

x + 2 = 8

Subtract 2 from both sides

x + 2 – 2 = 8 – 2

x = 6

Final solution:

(6, 2)

Check: Plug both values into one of the original equations.

x + y = 8

6 + 2 = 8

8 = 8

**Example 2:**

Solve the following system of equations by substitution.

y = 3x – 5

y = – 4x + 9

**Solution:**

Since both equations are already solved for* y*, we can substitute 3x – 5 for *y* in the second equation and then solve for the remaining variable, *x*.

3x-5 = -4x + 9

7x – 5= 9

7x=14

x=2

Now find *y* by substituting x = 2 into the first equation.

y = 3x-5

y= 3(2) -5

y=6-5

y=1

Thus, we have x = 2 and y = 1, or the ordered pair (2, 1). A check will show that the solution to the system of equations is (2, 1).

Final solution:

(2, 1)

Check: Plug both values into one of the original equations.

*y* = 3*x* – 5

1 = 3(2) – 5

1 = 1

### Compare Graphing and Substitution Methods

**Example 3:**

The sum of two numbers is 12. Twice the second number is three less than the second. Find the two numbers.

**Solution:**

Let *x* be the first number and* y* be the second number.

*x* + *y* = 12

2*y* = *x *– 3

**Method 1:**

Solve the system of equations by graphing.

Check your answer:

*x* + *y* = 12

9 + 3 = 12 ……….

2*y* = *x *– 3

2(3) = 9 – 3

6 = 6 ……….

The first number is 9, and the second number is 3.

**Method 2:**

Solve the system of equations by substitution.

**Step 1:** Solve one of the equations for either *x* or *y*.

*x* + *y* = 12

*x* + *y – y *= 12 – *y*

*x* = 12 – *y*

**Step 2:** Substitute for *x *and solve for* y*.

2*y* = *x *– 3

2*y* = (12 – *y*)– 3 …………

2y = 9 – *y*

*y* = 3

**Step 3:** Substitute 3 for *y* in one of the equations and solve for *x*.

*x* + *y* = 12

*x* + 3 = 12

*x* = 12 – 3

*x* = 9

**Step 4:** Check by substituting the values for *x *and *y* into each of the original equations.

**Example 4:**

Three hundred tickets were sold for the annual pancake breakfast. Adult tickets cost $5.00, and tickets for children under the age of twelve cost $2.50. Total receipts for the breakfast were $1187.50. Find the number of adult tickets sold and the number of children’s tickets sold.

**Solution:**

Let *x* be the number of adult tickets and* y *be the number of children’s tickets.

*x* + *y* = 300

5*x* + 2.5*y* = 1187.50

**Method 1**

Solve the system of equations by graphing.

Check your answer:

*x* + *y* = 300

175 + 125 = 300………….

5*x* + 2.5*y* = 1187.50

5(175) + 2.5(125) = 1187.50

875 + 312.5 = 1187.50

1187.5 = 1187.50…………

175 adult tickets were sold, and 125 children’s tickets were sold.

**Method 2**

Solve the system of equations by substitution.

**Step 1:** Solve one of the equations for either *x* or *y*.

*x* + *y* = 300

*x* + *y – y *= 300 – *y*

*x* = 300 – *y*

**Step 2:** Substitute for *x* and solve for* y*.

5*x* + 2.5*y* = 1187.50

5(300 – *y*) + 2.5*y* = 1187.50……………

– 2.5y = – 312.5

*y* = 125

**Step 3:** Substitute 125 for* y* in one of the equations and solve for* x*.

*x* + *y* = 300

*x* + 125 = 300

*x* = 300 – 125

*x* = 175

**Step 4:** Check by substituting the values for *x* and* y* into each of the original equations.

#### Exercise

- In algebra, _____________________ means putting numbers where the letters are.
- Use substitution to solve the following system of equations.

x = y + 6

x + y =10

3. Use substitution to solve the following system of equations.

y = 2x – 1

2x + 3y = –7

4. When is using a graph to solve a system of equations more useful than the substitution method?

5. Use substitution to solve the following system of equations.

y = 6 – x

4x – 3y = –4

6. Use substitution to solve the following system of equations.

x = –y + 3

3x – 2y = –1

7. The difference between the two numbers is 14. When twice the first number is added to the second number, the result is 10. Find the two numbers.

8. The sum of two numbers is –16. One number is three times the other number. Find the two numbers.

9. The perimeter of a rectangle is 78 centimeters. The length of the rectangle is 6 centimeters less than twice the width. Find the dimensions of the rectangle.

10. Maggie and Addison went shopping after Christmas. Maggie bought 4 pairs of socks and 2 sweaters and spent $60. Addison spent $75 on 1 pair of socks and 3 sweaters. Find the cost of one pair of socks. Find the cost of one sweater.

#### Concept Map:

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