Start with the meaning
Begin with one-step equations, negative numbers and collecting like terms. You should know that 3x means 3 × x, not x + 3. If these are unfamiliar, use the linked simpler guide first. NCERT connection: the linked Exemplar resource covers this topic. These are original BananaLearn explanations and practice, not official NCERT exercise solutions. Exemplar and current textbook chapter sequences can differ; match the skill to your school’s book rather than relying on a chapter number.
Equality is a relationship
The equation 4x + 5 = 2x + 17 says that the two expressions have the same value for the solution. Subtracting 2x from both sides leaves 2x + 5 = 17. Subtracting 5 leaves 2x = 12, and dividing both sides by 2 gives x = 6.
The equal sign is not an instruction to write the next calculation. Each line should remain a true equation for the same solution. Write one operation per line until you can explain every step without skipping.
Why terms appear to change signs
From 5x + 7 = 3x + 19, subtract 3x on both sides. Then subtract 7 on both sides. The shorter written form 5x − 3x = 19 − 7 comes from those two balanced operations; no term physically jumps across the equal sign.
You can instead collect x on the right. That may produce negative coefficients, but it can still be correct. Choose a route you can track clearly, rather than assuming only one side is allowed to contain the variable.
Expand brackets before collecting
For 3(x + 2), multiply both terms inside the bracket: 3x + 6. A minus sign also applies to the whole bracket: −(x − 4) becomes −x + 4. Leaving one term unchanged creates a different expression and therefore a different equation.
After expanding, combine like terms on each side before choosing your balance operations. When fractions occur, multiplying every term on both sides by a common non-zero denominator can remove them. Do not multiply just the term that looks inconvenient.
Check the original equation
Substitute your answer into both original expressions separately. For x = 6, 4x + 5 gives 29 and 2x + 17 also gives 29. Checking only the final line x = 6 does not test an earlier sign or bracket mistake.
Sometimes the variable terms cancel. If you reach 2 = 5, the original equation has no solution. If you reach 2 = 2, every allowed value satisfies it. In the simple examples here the variable may be any real number; there is no variable denominator imposing restrictions.
Worked examples, step by step
Example 1: Solve 5x + 4 = 2x + 25
- Subtract 2x from both sides: 3x + 4 = 25.
- Subtract 4: 3x = 21. Divide both sides by 3: x = 7.
- Check the original sides: 5 × 7 + 4 = 39 and 2 × 7 + 25 = 39.
Example 2: Solve 3(x + 2) = x + 18
- Expand the left side: 3x + 6 = x + 18.
- Subtract x and then subtract 6: 2x = 12. Divide by 2 to get x = 6.
- Check: 3(6 + 2) = 24 and 6 + 18 = 24. Both sides match.
Example 3: A rectangle’s length is x + 5 cm and its width is x cm. Its perimeter is 38 cm. Find the dimensions.
- Perimeter counts two lengths and two widths: 2(x + 5) + 2x = 38.
- Expand and collect: 4x + 10 = 38. Subtract 10, then divide by 4: x = 7.
- Width is 7 cm; length is 12 cm. Check the context: 2 × 12 + 2 × 7 = 38 cm, and both lengths are positive.
Common mistakes and useful repairs
Changing a sign without knowing the operation
Write the operation under both sides. From x − 6 = 10, adding 6 to each side gives x = 16. Saying only “move 6” can hide whether you are adding or subtracting.
Dividing only one term
From 2x + 6 = 18, dividing the entire equation by 2 gives x + 3 = 9. Writing x + 6 = 9 divides some terms but not others and does not preserve equality.
Ignoring what the variable means
An algebraic solution must also fit the story. A negative length or a non-whole count of people may be invalid even if an equation was solved correctly. Label x before forming a word-problem equation.
Try before revealing
Quick understanding check
Read the check answers
2x + 5 = 17. Subtracting 2x removes it from the right and changes 4x to 2x on the left.
2x − 6 = 10. The multiplier applies to both terms inside the bracket.
Grade 8 linear equations: variables on both sides — practice sheet
Work on paper before opening an answer. Print this question sheet with or without a separate worked key; no account is needed.
Name: ____________________ Date: __________
Solve 4x + 3 = x + 18.
Show answer and reasoning
x = 5. Subtract x: 3x + 3 = 18. Subtract 3 and divide by 3. Check: both original sides equal 23.
Solve 7x − 5 = 3x + 19.
Show answer and reasoning
x = 6. Subtract 3x and add 5: 4x = 24. Divide by 4. Both original sides equal 37.
Solve 2(x + 4) = x + 15.
Show answer and reasoning
x = 7. Expand: 2x + 8 = x + 15. Subtract x and 8. Check: 2 × 11 = 22 and 7 + 15 = 22.
Solve 3x + 8 = 5x + 2.
Show answer and reasoning
x = 3. Subtract 3x and then subtract 2: 6 = 2x. Divide by 2. Both original sides equal 17.
Solve 5 − 2x = x + 14.
Show answer and reasoning
x = −3. Subtract x and 5: −3x = 9, so x = −3. Check: 5 − 2(−3) = 11 and −3 + 14 = 11.
A rectangle has width x cm and length x + 3 cm. Its perimeter is 30 cm. Find both dimensions.
Show answer and reasoning
Width 6 cm; length 9 cm. 2x + 2(x + 3) = 30 gives 4x + 6 = 30, then x = 6. Check: 12 + 18 = 30 cm.
What happens when you solve 2x + 3 = 2x + 8?
Show answer and reasoning
No solution. Subtract 2x from both sides to get 3 = 8, which is false for every x.
What happens when you solve 2(x + 3) = 2x + 6?
Show answer and reasoning
Every real value of x works. Expansion gives 2x + 6 = 2x + 6. The two expressions are identical, so there is no single unique solution.
Grade 8 linear equations: variables on both sides — worked answer key
x = 5. Subtract x: 3x + 3 = 18. Subtract 3 and divide by 3. Check: both original sides equal 23.
x = 6. Subtract 3x and add 5: 4x = 24. Divide by 4. Both original sides equal 37.
x = 7. Expand: 2x + 8 = x + 15. Subtract x and 8. Check: 2 × 11 = 22 and 7 + 15 = 22.
x = 3. Subtract 3x and then subtract 2: 6 = 2x. Divide by 2. Both original sides equal 17.
x = −3. Subtract x and 5: −3x = 9, so x = −3. Check: 5 − 2(−3) = 11 and −3 + 14 = 11.
Width 6 cm; length 9 cm. 2x + 2(x + 3) = 30 gives 4x + 6 = 30, then x = 6. Check: 12 + 18 = 30 cm.
No solution. Subtract 2x from both sides to get 3 = 8, which is false for every x.
Every real value of x works. Expansion gives 2x + 6 = 2x + 6. The two expressions are identical, so there is no single unique solution.
Choose the next practice step
Use the questions in this guide to practise this specific skill. If you need another explanation, return to One-step equations: keep both sides equal before trying a fresh problem.
Help at home: Ask for the operation, not the answer
Cover the next line of a worked example. Ask your child which operation they would apply to both sides and why. If they say “move x”, ask them to name the addition or subtraction that makes that shorthand valid.
For a wrong answer, substitute it into the original equation together. Different left and right values show that something changed along the way. Compare consecutive lines to locate the first unequal step, then give a fresh equation with different numbers.
Keep one example of your child’s own working. If the same confusion returns across fresh questions, share that attempt with their teacher and ask which next step fits their current learning.
Questions students and parents ask
Can x be on the right at the end?
Yes. 6 = x and x = 6 mean the same thing. Equality works in both directions; choose the layout that makes your steps easiest to read.
Should I always move the smaller x term?
It often keeps the coefficient positive, which can simplify the arithmetic. It is a useful choice, not a compulsory rule. A correctly handled negative coefficient gives the same solution.
Are these solutions to the whole NCERT chapter?
No. This guide teaches selected skills using original questions. The official Grade 8 Exemplar link is a scope reference, and its chapter numbering may differ from your current school textbook.
About this resource
BananaLearn created these explanations, diagrams and practice questions. They are not reproduced textbook or past-exam questions. School sequences differ; use the prerequisites to choose a starting point. Read how we prepare learning guides.
Curriculum and teaching references used to check scope:
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