Mathematics Year 7: Solving first-degree equations and problems
Mathematics, Year 7 (Portugal): Solving means undoing operations on both sides so the solution never changes, and then checking that the answer makes sense in the story.
Undo the operations, on both sides
In 2.5x − 4 = 6 the unknown was multiplied by 2.5 and then 4 was taken away, giving 6. Undo in reverse order: 6 + 4 = 10, then 10 ÷ 2.5 = 4. In equation form it is the same rule as a balance: whatever you do to one side, do to the other. 2.5x − 4 + 4 = 6 + 4 gives 2.5x = 10, and dividing both sides by 2.5 gives x = 4. Check: 2.5 × 4 − 4 = 6. The solution need not be whole: 2x + 1 = 6 gives 2x = 5, so x = 5/2 = 2.5.
Why the steps keep the solution
Two equations are equivalent when they have exactly the same solutions. Adding or subtracting the same number on both sides, or multiplying or dividing both sides by a number other than zero, gives an equivalent equation, because you can always go back with the inverse operation. To justify that 3x + 6 = 21 and 3x = 15 are equivalent: the second comes from the first by subtracting 6 from both sides, and you return by adding 6. Then 3x = 15 and x = 5 are equivalent by dividing by 3. But 3x = 15 and 3x = 12 are not: their solutions are 5 and 4.
A problem, and does the answer fit?
Marta has 25 euros and saves 3 euros a week; Tomás has 7 euros and saves 6 euros a week. After how many weeks do they have the same amount? With x weeks: 25 + 3x = 7 + 6x. Subtract 3x from both sides: 25 = 7 + 3x. Subtract 7: 18 = 3x. Divide by 3: x = 6. Check: 25 + 3 × 6 = 43 and 7 + 6 × 6 = 43. Does 6 fit the story? Weeks are counted in whole numbers and 6 is one, so yes. Had the solution been 4.5, there would be no week when the amounts are equal, and it would be time to reread the problem.
Moving a term without changing its sign
Solve 4x + 9 = x + 30. A common slip: 4x − x = 30 + 9, so 3x = 39 and x = 13. It feels like "moving the 9 across", but 9 was added, so to remove it you subtract 9 from both sides, and it becomes 30 − 9. Test 13: 4 × 13 + 9 = 61, but 13 + 30 = 43, not equal. Correct: 4x − x = 30 − 9, so 3x = 21 and x = 7. Test: 4 × 7 + 9 = 37 and 7 + 30 = 37. Always substitute the answer into the original equation, not into one you already changed.
Comparing two plans
Two bike-rental shops: A charges 10 euros fixed plus 2 euros per hour, B charges 4 euros fixed plus 5 euros per hour. When do they cost the same? 10 + 2x = 4 + 5x. Subtract 2x: 10 = 4 + 3x. Subtract 4: 6 = 3x, so x = 2. Check: 10 + 2 × 2 = 14 and 4 + 5 × 2 = 14. For less than 2 hours B is cheaper, for more than 2 hours A is. Comparing plans, sharing costs, mixing quantities: whenever two ways of counting must agree, an equation finds the moment where they do.
Test what you learned
Which equation is equivalent to 5x − 4 = 21?
- a) 5x = 17
- b) x − 4 = 4,2
- c) 5x = 21
- d) 5x = 25
Correct answer: d) 5x = 25 — Right: adding 4 to both sides gives 5x = 25, and you can go back by subtracting 4. Both have solution 5: 5 × 5 − 4 = 21.
What is the solution of 7x − 9 = 3x + 11?
- a) x = 2
- b) x = 5
- c) x = 20
- d) x = 1/2
Correct answer: b) x = 5 — Right: 7x − 3x = 11 + 9, so 4x = 20 and x = 5. Check: 7 × 5 − 9 = 26 and 3 × 5 + 11 = 26.
Rita solved 3x + 8 = x + 20 like this: 3x − x = 20 + 8; 2x = 28; x = 14. But 14 is not the solution. What went wrong?
- a) She should have divided 28 by 3, the larger coefficient
- b) An equation cannot have x on both sides
- c) She added 8 to the right side; it should be 20 − 8 = 12
- d) 3x − x is 4x, not 2x
Correct answer: c) She added 8 to the right side; it should be 20 − 8 = 12 — Right: the 8 was added on the left, so you subtract 8 from both sides. Then 2x = 12 and x = 6. Check: 3 × 6 + 8 = 26 and 6 + 20 = 26.
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