Solving One-Step and Two-Step Equations Quiz — Secondary 2

Answer: 5

An equation is a balance: whatever you do to one side, you must do to the other, or the scale tips. Here 7 is being added to x, so you undo it by subtracting 7 from both sides. On the left, x + 7 − 7 leaves x alone. On the right, 12 − 7 = 5. So x = 5. Always check by putting the value back in the original equation: 5 + 7 = 12. True, so the answer is confirmed.

Answer: 7

The notation 3x means 3 multiplied by x. The opposite of multiplying by 3 is dividing by 3, so divide both sides by 3. On the left, 3x ÷ 3 = x. On the right, 21 ÷ 3 = 7. So x = 7. Check: 3 × 7 = 21. Correct. A frequent slip is to subtract 3 instead of dividing — but 3 is not added to x, it multiplies it, and you must undo the operation that is actually there.

Answer: 36

Here x is being divided by 4, so undo it by multiplying both sides by 4. On the left, (x/4) × 4 = x. On the right, 9 × 4 = 36. So x = 36. Check: 36/4 = 9. Correct. Notice the answer is bigger than 9 — that makes sense, because a number that shrinks to 9 after being cut in four must have been substantially larger to begin with.

Answer: 6

With two operations you undo them in reverse order of the priority rules: deal with the addition first, the multiplication second. Subtract 5 from both sides: 2x + 5 − 5 = 17 − 5, which gives 2x = 12. Now divide both sides by 2: x = 6. Check in the original: 2 × 6 + 5 = 12 + 5 = 17. Correct. The classic mistake is dividing by 2 first, because then you must also divide the 5 and the 17, which is more work and easy to botch.

Answer: 5

Three is being subtracted, so the first move is to add 3 to both sides: 5x − 3 + 3 = 22 + 3, giving 5x = 25. Then divide both sides by 5: x = 5. Check: 5 × 5 − 3 = 25 − 3 = 22. Correct. Keep in mind that the sign in front of a number tells you what to do to remove it: a minus sign is undone by adding.

Answer: x = 18

Take the addition off first: subtract 4 from both sides to get x/3 = 6. Now x is divided by 3, so multiply both sides by 3: x = 18. Check in the original equation: 18/3 + 4 = 6 + 4 = 10. Correct. The wrong answer 42 comes from adding the 4 instead of subtracting it: that gives x/3 = 14 and then x = 42. Ask yourself each time what the number is doing to the unknown, and undo exactly that — here 4 is being added, so it must be subtracted.

Answer: 3

Subtract 9 from both sides first: −2x + 9 − 9 = 3 − 9, which gives −2x = −6. Now divide both sides by −2. A negative divided by a negative is positive: −6 ÷ −2 = 3, so x = 3. Check in the original: −2 × 3 + 9 = −6 + 9 = 3. Correct. The step people rush is the sign: if you divide −6 by 2 instead of by −2 you get −3, which does not work in the equation. The coefficient you divide by is the whole thing in front of x, minus sign included.

Answer: x = 4

Rewrite the left side to see the coefficient clearly: 7 − x is the same as −x + 7. Subtract 7 from both sides: −x = 3 − 7 = −4. You now have −x, not x, so multiply (or divide) both sides by −1: x = 4. Check in the original: 7 − 4 = 3. Correct. The answer −4 is the trap: it is the value of −x, not of x, and stopping one line too early is what produces it.

Answer: 7

Two routes work, and both must land on the same number. Route one: divide both sides by 4 first, since the whole bracket is multiplied by 4. That gives x − 2 = 5, then add 2 to get x = 7. Route two: distribute first. 4(x − 2) = 4x − 8, so 4x − 8 = 20, then 4x = 28 and x = 7. Check: 4(7 − 2) = 4 × 5 = 20. Correct. When the right-hand side divides evenly by the factor outside, route one is quicker; when it does not, distribute instead. The error to avoid in route two is forgetting to multiply the −2 by 4.

Answer: 4 cm

The perimeter of a rectangle is twice the width plus twice the length, so P = 2x + 2(x + 5). Distribute: 2x + 2x + 10 = 4x + 10. Set that equal to 26: 4x + 10 = 26. Subtract 10: 4x = 16. Divide by 4: x = 4. So the width is 4 cm and the length is 9 cm. Check the perimeter: 4 + 9 + 4 + 9 = 26 cm. Correct. The frequent mistake here is writing x + (x + 5) = 26, which is the half-perimeter, and it would give x = 10.5 — a length of 15.5 cm, far too big for a 26 cm perimeter.

Answer: 3 kg

Translate the picture into an equation. On the left pan there are 3 blocks and 2 discs, so its mass is 3x + 2. On the right pan there are 11 discs, so 11. Balanced means equal: 3x + 2 = 11. Remove 2 discs from each pan — that is subtracting 2 from both sides — and the scale stays balanced: 3x = 9. Now the three blocks together weigh 9 kg, so one block weighs 9 ÷ 3 = 3 kg. Check: 3 blocks at 3 kg plus 2 discs gives 9 + 2 = 11 kg, which matches the right pan. This picture is exactly what the balance method means: every legal algebra step is a move that keeps the scale level.

Answer: 7 km

Name the unknown: let x be the number of kilometres. The flat fee is paid once, so it is a constant: 4. The distance charge depends on x, so it is 2x. The total is 2x + 4, and you are told the total is 18. The equation is 2x + 4 = 18. Subtract 4 from both sides: 2x = 14. Divide by 2: x = 7 kilometres. Check: 4 + 2 × 7 = 4 + 14 = 18 dollars. Correct. Recognising which number is a one-time constant and which one multiplies the variable is the whole skill in this type of problem, and it is the same structure you will meet again with linear functions.