SOLVING SYSTEMS OF EQUATIONS BY SUBSTITUTION

The following steps will be useful to solve system of equations 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. 

Solved Examples

Example 1 :

Solve the following system of equations by substitution. 

x - 5y + 17  =  0  and 2x + y + 1  =  0  

Solution :

The above explained steps have been illustrated in the picture shown below. 

Therefore, the solution is 

(x, y)  =  (-2, 3)

Example 2 :

Solve the following system of equations by substitution. 

5x - 3y - 8 = 0  and  2x - 3y - 5  = 0

Solution :

5x - 3y - 8  =  0 -----(1)

2x - 3y - 5  =  0 -----(2)

Step 1 :

In both (1) and (2), we have the same coefficient for y. 

So, solve (1) for 3y. 

5x - 3y - 8  =  0

-3y  =  8 - 5x

Multiply each side by (-1).

3y  =  5x - 8 -----(3)

Step 2 : 

Substitute (5x - 8) for y into (2). 

(2)-----> 2x - (5x - 8) - 5  =  0

2x - 5x + 8 - 5  =  0

-3x + 3  =  0

Subtract 3 from each side. 

-3x  =  -3

Divide each side by (-3).

x  =  1

Step 3 :

Substitute 1 for x into (3).

(3)-----> 3y  =  5(1) - 8

3y  =  -3

Divide each side by 3.

y  =  -1

Therefore, the solution is 

(x, y)  =  (1, -1)

Example 3 :

Solve the following system of equations by substitution. 

y  =  6x - 11 and -2x - 3y  =  -7

Solution :

y  =  6x - 11 -----(1)

-2x - 3y  =  -7 -----(2)

Step 1 :

In the given two equations, already (1) is solved for y. 

So, we don't have to do anything more in this step. 

Step 2 : 

Substitute (6x - 11) for y into (2). 

(2)-----> -2x - 3(6x - 11)  =  -7

-2x - 18x + 33  =  -7

-20x + 33  =  -7

Subtract 33 from each side. 

-20x  =  -40

Divide each side by (-20).

x  =  2

Step 3 :

Substitute 2 for x into (1).

(1)-----> y  =  6(2) - 11

y  =  12 - 11

y  =  1

Therefore, the solution is 

(x, y)  =  (2, 1)

Example 4 :

Solve the following system of equations by substitution. 

2x - 3y  =  -1 and y  =  x - 1

Solution :

2x - 3y  =  -1 -----(1)

y  =  x - 1 -----(2)

Step 1 :

In the given two equations, already (2) is solved for y. 

So, we don't have to do anything more in this step. 

Step 2 : 

Substitute (x - 1) for y into (1). 

(2)-----> 2x - 3(x - 1)  =  -1

2x - 3x + 3  =  -1

-x + 3  =  -1

Subtract 3 from each side. 

-x  =  -4

Multiply each side by (-1).

x  =  4

Step 3 :

Substitute 4 for x into (2).

(2)-----> y  =  4 - 1

y  =  3

Therefore, the solution is 

(x, y)  =  (4, 3)

Example 5 :

Solve the following equations by substitution method 

y  =  -3x + 5 and 5x - 4y  =  -3

Solution :

y  =  -3x + 5 -----(1)

5x - 4y  =  -3 -----(2)

Step 1 :

In the given two equations, already (1) is solved for y. 

So, we don't have to do anything more in this step. 

Step 2 : 

Substitute (-3x + 5) for y into (2). 

(2)-----> 5x - 4(-3x + 5)  =  -3

5x + 12x - 20  =  -3

17x - 20  =  -3

Add 20 to each side. 

17x  =  17

Divide each side by 17.

x  =  1

Step 3 :

Substitute 1 for x into (1).

(1)-----> y  =  -3(1) + 5

y  =  -3 + 5

y  =  2

Therefore, the solution is 

(x, y)  =  (1, 2)

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