The Lesson
Simultaneous equations are a set of several equations with several unknowns. We can use the substitution method to find the values of the unknowns which solve both equations at the same time. An unknown from one equation is substituted into the other equation, allowing one unknown to be found in one equation.
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.
We can substitute Equation (1) into Equation (2).
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
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
x − y = 1
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