    ### Simultaneous Equations by Substitution

Simultaneous equations are two equations with two unknown variables, and we want to find the same solution

There are 5 steps

In steps b and d we use a combination of Solving Equations − Add and Subtract and Solving Equations − Multiply and Divide
Please look at the above before continuing

• #### Example 1.   Solve the following two equations by substitution                   2x + y = 5                   3x − y = 15

• i. Label each equation

 Equation Label 2x + y = 5 (1) 3x − y = 15 (2)

ii. Rearrange equation (1) to make y the subject

 2x + y = 5 Note (3) − 2x = − 2x
 y = 5 − 2x

Note: (3) inverse of + 2x is − 2x

iii. Substitute y = 5 − 2x into equation (2):

 3x − y = 15 3x − (5 − 2x) = 15 Note (4) 3x − 5  +  2x = 15

Note: (4) − ( − 2) = + 2

iv. Rearrange to find the value of x

 3x − 5 + 2x = 15 Note (5) + 5 = + 5
 5x = 20 Note (6) ÷ 5 = ÷ 5
 x = 4

Note: (5) inverse of − 5 is + 5
Note: (6) inverse of × 5 is ÷ 5

v. put x = 4 into equation:

So x = 4 and y = −3

• #### Example 2.    Solve the following two equations by substitution                   4a + b = 20                   5a + 3b = 32

• i. Label each equation

 Equation Label 4a + b = 20 (1) 5a + 3b = 32 (2)

ii. Rearrange equation (1) to make b the subject

 4a + b = 20 Note (3) − 4a = − 4a
 b = 20 − 4a

Note: (3) inverse of + 4a is − 4a

iii. Substitute b = 20 − 4a into equation (2):

 5a + 3(b) = 32 5a + 3(20 − 4a) = 32 5a + 60  −  12a = 32

iv. Rearrange to find the value of a

 5a + 60 − 12a = 32 Note − 60 = − 60 (4)
 −7a = − 28 Note ÷ −7 = ÷ −7 (5)
 a = 4

Note: (4) inverse of + 60 is − 60
Note: (5) inverse of × −7 is ÷ −7

v. put a = 4 into equation:

So a = 4 and b = 4 to:  