#### Introduction:

## Solving System of Equations by Elimination:

The addition (or elimination) 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 System of Equations by the Addition Method:

- If necessary, rewrite each equation in standard form, ax + by = c.
- If necessary, multiply one or both equations by a constant(s) so that when the equations are added, the sum will contain only one variable.
- Add the respective sides of the equations. This will result in a single equation containing only one variable.
- Solve the equation obtained in step 3.
- Substitute the value found in Step 4 into either of the original equations. Solve that equation to find the value of the remaining variable.
- Write the solution as an ordered pair.
- Check your solution in all equations in the system.

### Solve a System of Equations by Adding

**Example 1:**

What is the solution to the system of equations?

2*x* + *y *= 11

*x* + 3*y* = 18

**Solution:**

Since adding the equations at this point would not eliminate a variable, the first equation is multiplied by –2.

Now solve for x by substituting 5 for y in either of the original equations.

The solution is (3, 5).

**Example 2:**

What is the solution to the system of equations?

2*x* + 5*y *= 3

3*x* – 5*y* = 17

**Solution:**

2*x* + 5*y *= 3

3*x* – 5*y* = 17

Adding the equations yields:

The *x*-variable can now be obtained using the same steps used for the substitution method.

2*x* + 5*y *= 3

3*x* – 5*y* = 17

Substitute *x* = 2 into either equation:

2*x* + 5*y *= 3

2(4) + 5y= 3

8 + 5*y *= 3

5*y *= -5

*y* = -1

The solution of the system is (4, -1).

### Understand Equivalent Systems of Equations

**Example 3:**

What is the solution to the system of equations?

3*x *– 2*y *= 8

4x + 10y = -2

**Solution:**

Before adding equations, multiply each side of one of the equations by a constant that makes either the x or y terms opposites.

3*x *– 2*y *= 8

4x + 10y = -2

Multiply the top equation by 5:

5(3x – 2y) = 5(8)

Now substitute 2 for y in either of the two equations in the system.

2 (2) + 5y = -1

5y = -5

y = -1

The solution is (2, -1).

### Apply Elimination

**Example 4:**

Ashley wants to use milk and orange juice to increase the amount of calcium and vitamin A in her daily diet. An ounce of milk contains 37 milligrams of calcium and 57 micrograms of vitamin A. An ounce of orange juice contains 5 milligrams of calcium and 65 micrograms of vitamin A. How many ounces of milk and orange juice should Ashley drink each day to provide exactly 500 milligrams of calcium and 1,200 micrograms of vitamin A?

**Solution:**

**Formulate:**

Let *x *= number of ounces of milk

Let *y *= number of ounces of orange juice

Calcium: 37*x *+ 5*y *= 500

Vitamin A: 57*x *+ 65*y *= 1,200

**Compute:**

Multiply each equation by constants to eliminate one variable.

37*x *+ 5*y *= 500……………….Multiply by –13

-481*x *– 65*y *= -6,500

-481*x *– 65*y *= -6,500

57*x *+ 65*y *= 1,200

___________________________

-424*x *= -5,300

*x *= 12.5

Solve for y:

37(12.5) + 5*y *= 500

5*y *= 37.5

*y *= 7.5

The solution is (12.5, 7.5)

**Interpret:**

Drinking 12.5 ounces of milk and 7.5 ounces of orange juice each day will provide Ashley with the required amounts of calcium and vitamin A.

### Choose a Method of Solving

**Example 5:**

What is the solution to the system of equations?

A.

3*x *– 2*y *= 38

*x* = 6 – *y*

**Solution:**

Since the equation is already solved for one of the variables, you can easily substitute 6 – *y* for *x*.

3(6 – *y*)- 2*y *= 38

18 – 3y – 2y = 38

18 – 5y = 38

–5y = 38 – 18

–5y = 20

y = 20/–5

y = –4

Solve for x,

*x* = 6 – (–4)

*x* = 10

The solution is (10, –4)

B.

6*x *+ 12*y *= –6

3*x *– 2*y* = –27

**Solution:**

The coefficient of *y* in the second equation is an integer multiple of the coefficient of *y* in the first equation. This makes it easy to eliminate the *y* variable.

6*x *+ 12*y *= –6

3*x *– 2*y* = –27………..Multiply by 6 18*x *– 12*y* = –162

Add the equation and solve for *x,*

6*x *+ 12*y *= –6

18*x *– 12*y* = –162

_____________________

24*x* = –168

*x* = –7

Now solve for *y*

3(–7)– 2*y* = –27

–21 –2*y* = –27

–2*y* = –27 + 21

*y* = 3

#### Exercise:

- What is the solution to the system of equations?

2*x* – 4*y *= 2

–*x* + 4*y* = 3

2. What is the solution to the system of equations?

2*x* + 3*y *= 1

–2*x* + 2*y* = –6

3. What is the solution to the system of equations?

*x* + 2*y *= 4

2*x* – 5*y* = –1

4. What is the solution to the system of equations?

2*x* + *y *= 2

*x* – 2*y* = –5

5. A florist is making regular bouquets and mini bouquets. The florist has 85 roses and 163 peonies to use in the bouquets. How many of each type of bouquets can the florist make?

6. What is the solution to the system of equations? Explain your choice of the solution method.

6*x* + 12*y *= –6

3*x* – 2*y* = –27

7. What is the solution to the system of equations? Explain your choice of the solution method.

3*x* – 2*y* = 38

*x* = 6 – *y*

8. The sum of 5 times the width of a rectangle and twice its length is 26 units. The difference of 15 times the width and three times the length is 6 units. Write and solve a system of equations to find the length and width of the rectangle.

9. Ella is a landscape photographer. One weekend at her gallery she sells a total of 52 prints for a total of $2,975. How many of each size print did Ella sell?

10. Is the given pair of systems of equations equivalent? Explain.

3*x* – 9*y* = 5

6*x* + 2*y* = 18

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