USE SUBSTITUTION TO SOLVE EACH SYSTEM OF EQUATIONS

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. 

Solving System of Equations by Substitution - Steps

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.  

Use Substitution to Solve Each System of Equations 

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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