**Use Substitution to Solve Each System of Equations :**

In this section, you will learn how to solve a system of linear equations with two unknowns using substitution.

**Step 1 : **

In the given two equations, solve one of the equations either for x or y.

**Step 2 : **

Substitute the result of step 1 into other equation and solve for the second variable.

**Step 3 : **

Using the result of step 2 and step 1, solve for the first variable.

**Problem 1 :**

Solve the following system of equations using substitution.

-4x + y = 6 and -5x - y = 21

**Solution :**

-4x + y = 6 -----(1)

-5x - y = 21 -----(2)

**Step 1 :**

Solve (1) for y.

-4x + y = 6

Add -4x to each side.

y = 6 + 4x -----(3)

**Step 2 : **

Substitute (6 + 4x) for y into (2).

(2)-----> -5x - (6 + 4x) = 21

-5x - 6 - 4x = 21

Simplify.

-9x - 6 = 21

Add 6 to each side.

-9x = 27

Divide each side (-9).

x = -3

**Step 3 :**

Substitute -3 for x into (3).

(3)-----> y = 6 + 4(-3)

y = 6 - 12

y = -6

Therefore, the solution is

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

**Problem 2 :**

Solve the following system of equations using substitution.

2x + y = 20 and 6x - 5y = 12

**Solution :**

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

6x - 5y = 12 -----(2)

**Step 1 :**

Solve (1) for y.

2x + y = 20

Subtract 2x to each side.

y = 20 - 2x -----(3)

**Step 2 : **

Substitute (20 - 2x) for y into (2).

(2)-----> 6x - 5(20 - 2x) = 12

6x - 100 + 10x = 12

Simplify.

16x - 100 = 12

Add 100 to each side.

16x = 112

Divide each side 16.

x = 7

**Step 3 :**

Substitute 7 for x into (3).

(3)-----> y = 20 - 2(7)

y = 20 - 14

y = 6

Therefore, the solution is

(x, y) = (7, 6)

**Problem 3 :**

Solve the following system of equations using substitution.

y = -2 and 4x - 3y = 18

**Solution :**

y = -2 -----(1)

4x - 3y = 18 -----(2)

From (1), substitute -2 for y into (2).

4x - 3(-2) = 18

4x + 6 = 18

Subtract by 6 from each side.

4x = 12

Divide each side by 4.

x = 3

Therefore, the solution is

(x, y) = (3, -2).

**Problem 4 : **

Solve the following system of equations using substitution.

2x + 3y = 5 and 3x + 4y = 7

**Solution : **

2x + 3y = 5 -----(1)

3x + 4y = 7 -----(2)

**Step 1 :**

Multiply (1) by 3.

(1) ⋅ 3 -----> 6x + 9y = 15

Solve for 6x.

6x = 15 - 9y -----(3)

**Step 2 :**

Multiply (2) by 2.

(2) ⋅ 2 -----> 6x + 8y = 14

From (3), substitute (15 - 9y) for 6x.

(15 - 9y) + 8y = 14

Simplify.

15 - 9y + 8y = 14

15 - y = 14

Subtract 15 from each side.

-y = -1

Multiply each side by (-1).

y = 1

**Step 3 : **

Substitute 1 for y into (3).

(3)-----> 6x = 15 - 9(1)

6x = 15 - 9

6x = 6

Divide each side by 6.

x = 1

Therefore, the solution is

(x, y) = (1, 1)

After having gone through the stuff given above, we hope that the students would have understood, how to solve system of linear equations by substitution method.

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