Algebra essentials, Year 8: more practice

2p − 3q

HintGroup the p terms together and the q terms together, keeping each sign with its term.

The why6p − 4p = 2p and 2q − 5q = −3q. The minus sign in front of a term travels with it when you rearrange.

5a² + 4a

Hinta² and a are different terms, so collect each kind separately.

The why3a² + 2a² = 5a² and 5a − a = 4a. The result cannot be shortened further because a² and a are unlike terms.

x = 6

HintDo the opposite of subtracting 9 to both sides.

The whyAdd 9 to both sides: x = −3 + 9 = 6. Check: 6 − 9 = −3.

35

HintDo the opposite of dividing by 5 to both sides.

The whyMultiply both sides by 5: x = 7 × 5 = 35. Check: 35 ÷ 5 = 7.

x = 5

HintUndo the subtraction first, then the multiplication.

The whyAdd 8 to both sides: 5x = 25. Divide by 5: x = 5. Check: 5 × 5 − 8 = 17.

x = 9

HintUndo the division first, then the addition.

The whyMultiply both sides by 2: x + 3 = 12. Subtract 3: x = 9. Check: (9 + 3) ÷ 2 = 6.

x = 4

HintTake away 3x from both sides so the x terms are all on one side.

The whySubtracting 3x gives 4x − 4 = 12, so 4x = 16 and x = 4. Check: 7 × 4 − 4 = 24 and 3 × 4 + 12 = 24.

x = (y − 5) ÷ 3

HintUndo the + 5 first, then undo the multiplication by 3.

The whySubtract 5 from both sides: y − 5 = 3x. Divide both sides by 3: x = (y − 5)/3. Check with x = 2: y = 11, and (11 − 5)/3 = 2.

x > 4

HintTreat it like an equation: add 2, then divide by 3.

The why3x > 12, so x > 4. The solutions are numbers bigger than 4, but not 4 itself, because 3 × 4 − 2 = 10 is not greater than 10.

(1, −1)

HintPut each x-value into the rule and compare the answer with the given y-value.

The whyFor x = 1, 2 × 1 − 3 = −1, so (1, −1) is on the line. For x = 2 the rule gives 1, not 3, and for x = 3 it gives 3, not 6.