How can I solve a two-step linear equation without skipping steps?
Quick answer
Write each equation transformation on its own line. For a two-step equation like 3x + 5 = 20, subtract 5 from both sides, then divide both sides by 3 to isolate x. Keep the equation balanced by applying each operation to both sides. Finally, substitute your answer into the original equation and simplify both sides to see whether they match. [S1] [S2]
The question, the details, the next step.
I can sometimes solve an equation in my head, but then I lose track of what I did. What should I write down for a two-step equation?
Give every operation its own line, and show what you do to both sides. For example, in 3x + 5 = 20, first remove the added 5 by subtracting 5 from both sides. Then divide both sides by 3 to make the coefficient of x equal to 1. [S1]
3x + 5 = 20 3x + 5 − 5 = 20 − 5 3x = 15 3x ÷ 3 = 15 ÷ 3 x = 5
The middle lines may feel repetitive, but they show that each transformation keeps the equation balanced. OpenStax’s example also shows subtracting the constant from both sides, then dividing both sides by the coefficient. [S1]
How do I know which operation to undo first, especially when there are negative numbers?
For an equation like 4x − 7 = 9, first cancel the subtraction of 7 by adding 7 to both sides. Then divide both sides by 4 to make the coefficient of x equal to 1:
4x − 7 = 9 4x − 7 + 7 = 9 + 7 4x = 16 4x ÷ 4 = 16 ÷ 4 x = 4
The minus sign belongs to the term: adding 7 cancels −7. Rather than moving a term across the equals sign without explanation, show the same operation on each side. For more complicated equations, simplify each side and collect variable terms and constants as needed before making the variable’s coefficient 1. [S2]
What does a substitution check look like? Should I use the equation after I solve it or the one I started with?
Use the original equation, before you changed it. Put your proposed value wherever the variable appears, then simplify each side separately. For 4x − 7 = 9 with x = 4:
Left side: 4(4) − 7 = 16 − 7 = 9 Right side: 9
Both sides equal 9, so the check works. If they do not match, compare your written steps with the original equation and look for a sign, arithmetic, or division error. Substituting the proposed solution into the original equation is a way to check whether it makes the equation true. [S2]
Can you give me a short routine to use on homework, plus a couple of problems to practise?
Use this routine: copy the equation carefully; simplify either side if needed; cancel the added or subtracted constant by doing the same operation on both sides; then divide or multiply both sides to make the variable’s coefficient 1. Write each result on a new line. Finally, substitute your answer into the original equation.
Try these before reading the worked answers:
- 2x + 6 = 18
Subtract 6 from both sides: 2x = 12. Divide both sides by 2: x = 6. Check: 2(6) + 6 = 18.
- 5x − 8 = 12
Add 8 to both sides: 5x = 20. Divide both sides by 5: x = 4. Check: 5(4) − 8 = 12.
For each check, compare the simplified left side with the right side of the original equation. These examples have one solution each; linear equations can also have no solution or have all real numbers as solutions. [S2]
A practical checklist
Copy the original equation, including every negative sign.
Simplify either side if needed.
Apply the same operation to both sides and write each new equation on its own line.
Make the variable’s coefficient 1.
Substitute the proposed value into the original equation and compare both sides.
A few more useful answers
Can I do two steps in my head if I know the answer?
You can, but writing one operation per line makes your reasoning easier to check and helps locate mistakes. For practice, show the steps even when the answer seems obvious.
What if substitution gives different values on the two sides?
The proposed value does not make the original equation true as written. Recheck the signs and arithmetic in each line, then solve again; check against the original equation.
Sources
References used for this guide.
- [S1] OpenStax College Algebra with Corequisite Support: Linear Equations in One Variable openstax.org · Checked
- [S2] OpenStax Elementary Algebra: General Strategy for Solving Linear Equations openstax.org · Checked