Simultaneous Equations: Equal Values and Substitution Method
Simultaneous equations are two equations that share one solution. To solve, the equal values method rearranges both to y = x − 1 and y = 8 − 2x, then sets them equal. The substitution method puts x = y + 1 into the other equation.
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Equal Values Method
- Make the same variable (like y) the subject in both equations.
- Set the two expressions equal and solve for x.
- Substitute the x value back to find y.
Substitution Method
- Rearrange one equation to express one variable in terms of the other, for example .
- Substitute this into the other equation so it has only one variable, then solve.
- Exam tip: always substitute both values back into the original equations to check.
Choosing the Right Method
- Use equal values when the same variable is easy to make the subject.
- Use substitution when one equation is easy to rearrange to remove one variable.
- Both methods give the same solution for x and y.
Practice Questions
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We have the following simultaneous equations. Do , satisfy this simultaneous equation?
Correct! 🎉 +10 pointsNot quite right
Substituting and into both equations shows that both equations are satisfied: and . Therefore, and is the solution to the simultaneous equations.
We have the following simultaneous equations. Do , satisfy this simultaneous equation?
Correct! 🎉 +10 pointsNot quite right
Substituting and gives , which is correct, but , which is not 8. Therefore, and do not satisfy the simultaneous equations.
Solve the simultaneous equations:
Correct! 🎉 +20 pointsNot quite right
Add the two equations together: , which simplifies to , so . Substituting into gives . Therefore, the solution is .
Solve the simultaneous equations:
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Set the two expressions for equal to each other: . Solving for gives . Substituting into gives . Therefore, and is the solution.
Solve the simultaneous equations:
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Substitute into the second equation: . This simplifies to , so and . Substituting back gives . Therefore, the solution is .
Solve the simultaneous equations:
Correct! 🎉 +30 pointsNot quite right
Rearrange the second equation to get . Substitute this into the first equation: . Simplifying gives , so and . Substituting back gives . Therefore, the solution is .
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Interactive Activity
Solve simultaneous equations step-by-step using equal values and substitution
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Students Also Ask
The questions students bump into most on this topic
Both solve the same simultaneous equations. The equal values method isolates the same variable in both equations and sets the two expressions equal. The substitution method isolates one variable in a single equation, then replaces it in the other. Each one reduces the pair to a single unknown.
You can choose whichever is simpler for your equations. If isolating the same variable in both equations is easy, the equal values method is quick. If one variable is already on its own, or easy to isolate, the substitution method tends to be faster.
Yes. Both methods solve the same pair of simultaneous equations, so they always reach the same solution. For x + y = 5 and x - y = 1, each method gives x = 3 and y = 2. Pick the method that suits the problem in front of you.
Because both expressions equal the same variable. If you isolate y in each equation and get y = 5 - x and y = x - 1, then 5 - x and x - 1 both equal y. So they equal each other, leaving one equation in one unknown.
No. It does not matter which equation or variable you select, so choose the one that is easiest to rearrange. A simpler starting point usually means less algebra and fewer mistakes. Whichever variable you pick, the substitution still leads to the same final solution.
Substitute your values back into both original equations. If the left-hand side equals the right-hand side in each one, your solution is correct. For x = 3 and y = 2, check 3 + 2 = 5 and 3 - 2 = 1. Both are true.