How to Solve Simultaneous Equations Using the Substitution Method
Solving simultaneous equations using the substitution method is easy.Question
Solve the simultaneous equations shown below using the substitution method.Step-by-Step:
1
Equation (1) tells us that y = 2x.
Substitute this value of y into Equation (2).
x + 2x = 3
2
Solve for x.
x = 1 is a solution to the simultaneous equations.
x + 2x = 3 | |
3x = 3 | Add the like x terms |
3x ÷ 3 = 3 ÷ 3 | Divide both sides by 3 |
x = 1 |
3
Substitute the variable we have just found (x = 1) into one of the equations.
Solve for y.
y = 2 is a solution to the simultaneous equations.
y = 2x | Substitute into Equation (1) |
y = 2( 1 ) | |
y = 2 × 1 | |
y = 2 |
Answer:
We have solved the simultaneous equations:x = 1, y = 2 solves
y = 2x
x + y = 3
A Real Example of How to Solve Simultaneous Equations Using the Substitution Method
The example above was simple. Equation (1) told us what "y = ". In the following example, we will have to find what "y = " by rearranging one of the equations. Note: you could find what "x = "... it doesn't matter which unknown you substitute.Question
Solve the simultaneous equations shown below using the substitution method.Step-by-Step:
1
Rearrange Equation (2) to find "y =".
We have rearranged x − y = 1 to find what "y = ".
y = x − 1
x − y = 1 | |
x − y + y = 1 + y | Add y to both sides |
x = 1 + y | |
x − 1 = 1 + y − 1 | Subtract 1 from both sides |
x − 1 = y | |
y = x − 1 |
2
Rearranged Equation (2) tells us that y = x − 1.
Substitute this value of y into Equation (1).
x + ( x − 1 ) = 5
3
Solve for x.
x = 3 is a solution to the simultaneous equations.
x + x − 1 = 5 | |
2x − 1 = 5 | Add the like x terms |
2x − 1 + 1 = 5 + 1 | Add 1 to both sides |
2x = 6 | |
2x ÷ 2 = 6 ÷ 2 | Divide both sides by 2 |
x = 3 |
4
Substitute the variable we have just found (x = 3) into one of the equations.
Solve for y.
y = 2 is a solution to the simultaneous equations.
x + y = 5 | Substitute into Equation (1) |
3 + y = 5 | |
3 + y − 3 = 5 − 3 | Subtract 3 from both sides |
y = 2 |
Answer:
We have solved the simultaneous equations:x = 3, y = 2 solves
x + y = 5
x − y = 1
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