How can I check a linear equation solution by substitution?
Quick answer
Replace the variable in the original equation with your proposed solution, then simplify the left and right sides separately. If they have the same value, the substitution makes a true statement and the proposed solution checks out. If they differ, recheck your solving and arithmetic. Testing the original equation checks the value against the equation you were asked to solve. [S1]
The question, the details, the next step.
I solved 3x + 5 = 20 and got x = 5. How do I check that without just looking back at my work?
Start again from the equation as it was originally written: 3x + 5 = 20. Substitute 5 wherever you see x, keeping the rest of the equation unchanged: 3(5) + 5 = 20. Now simplify the left side: 15 + 5 = 20, so 20 = 20. The two sides have the same value, so x = 5 satisfies the original equation. Substituting a proposed solution into the original equation and checking whether it becomes a true statement is the verification method. [S1]
The important part is to evaluate both sides, not just simplify one side and stop. Write the equality you get so you can see the comparison.
Why use the original equation? I already changed it to 3x = 15 before getting 5.
The rearranged line is useful for solving, but checking the original equation tests whether your answer works in the problem you started with. A mistake while rearranging could leave you with a later line that looks consistent even though it no longer represents the original equation. Check in the original form rather than merely repeating a transformed line. [S1]
For example, if you accidentally wrote 3x = 25, then found a value from that incorrect line, checking that value in 3x + 5 = 20 would reveal whether it actually satisfies the original equation. This check tells you whether the proposed value works; by itself, it does not show which earlier step caused a wrong answer.
What if substitution gives different numbers? Can you show me what to write?
Suppose you are checking x = 4 for 3x + 5 = 20. Substitute and simplify: 3(4) + 5 = 20, then 12 + 5 = 20, which becomes 17 = 20. Since the sides differ, x = 4 does not satisfy the original equation. That does not by itself tell you whether the issue was solving, substitution, or arithmetic, so retrace those steps.
A copyable layout is: Original equation: [write it here] Proposed value: x = [value] Substitute: [left side with the value] = [right side with the value] Simplify: [left-side value] = [right-side value] Conclusion: [equal, so it checks / unequal, so it does not check]
Can you give me a short routine I can use on the next equation?
Keep the original equation in front of you, replace every occurrence of the variable with the proposed value, and simplify the two sides separately. Compare their final values and state what the comparison means. For instance, with 2x - 7 = 9 and proposed x = 8, substitution gives 2(8) - 7 = 9, or 9 = 9, so the value checks.
Do not change the equation while performing the check: the goal is to test the proposed value in the original statement. The check establishes whether that value satisfies that equation; it does not independently prove that you solved every earlier step correctly or that no other solution exists.
A practical checklist
Copy the original equation before substituting.
Replace every occurrence of the variable with the proposed value, using parentheses when helpful.
Simplify the left and right sides separately.
Compare the final values: equal means the substitution checks; unequal means it does not.
If they differ, retrace the solving and arithmetic instead of changing the check to fit the answer.
A few more useful answers
Should I substitute into the original equation or the last line I solved?
Use the original equation for the final check. That tests the proposed value against the equation you were asked to solve, rather than only against a rearranged line. [S1]
If both sides are equal, does that prove there is only one solution?
It shows that the proposed value satisfies the equation. The substitution check alone does not establish whether other solutions exist.
What should I do if the sides do not match?
Check the substitution and arithmetic, then retrace your solving steps from the original equation. A mismatch shows that the proposed value does not satisfy it, but does not identify which step went wrong.
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra: General Strategy for Solving Linear Equations openstax.org · Checked