Algebra, Year 9: equations and substitution

45

HintDeal with the power before the multiplication, and remember what a negative times a negative gives.

The whySquaring comes first: (−3)² = (−3) × (−3) = 9. Then 5 × 9. The 5 is not squared, because the index belongs only to the x.

16; −16

HintAsk exactly what the small 2 is attached to in each case.

The whyAn index applies only to what it is written next to. In (−4)² that is the whole bracket, so the negative is squared too; in −4² it is just the 4, and the minus sign is applied afterwards.

14

HintPut the value in brackets everywhere the letter appears, then watch what subtracting a negative does.

The why(−2)² = 4 and 5 × (−2) = −10, so the expression is 4 − (−10) = 4 + 10. Writing the brackets in stops both sign slips at once.

−8

HintMultiply the three copies one pair at a time and track the sign after each step.

The why(−2) × (−2) = 4, and 4 × (−2) = −8. An even number of negatives multiplies to a positive and an odd number to a negative, so squares of negatives are positive but cubes are negative.

brackets

HintThey come first in the order of operations.

The whyWith x = −5, writing (−5)² makes the squaring cover the minus sign, giving 25. Typed as −5², a calculator squares the 5 first and gives −25.

20

HintUndo the addition first so the fraction stands alone, then undo the division.

The whySubtracting 3 from both sides leaves x/4 = 5, and multiplying both sides by 4 gives x = 20. Check: 20 ÷ 4 + 3 = 8.

x = 7

HintThe division was the last thing done to the left-hand side, so undo it first.

The whyMultiplying both sides by 3 gives x + 5 = 12, so x = 7. Here the whole of x + 5 is divided by 3, so the multiplication comes before the subtraction.

6 and 12, so x = 9

HintNothing on either side may escape the multiplication, including the terms with no fraction in them.

The whyMultiplying keeps an equation balanced only if every term is multiplied. Check in the original: 9 ÷ 3 + 2 = 5 and 9 − 4 = 5, so both sides agree.

x = 6

HintWork out what must have been taken away from ten, then ask what halves to give that.

The whyTen minus something is 7, so the something, x/2, is 3 and x = 6. Formally: subtract 10 from both sides to get −x/2 = −3, then multiply both sides by −2.

Into the original equation

HintGo right back to the line you were given, before any working.

The whyIf a slip happened when the fraction was cleared, every later line contains the same slip and would wrongly agree with your answer. Only the equation you started from is safe to check against.

4n − 7 = 25

HintFollow the sentence in order: what happens to the letter first, and what does it all come to?

The whyMultiplying n by 4 gives 4n, subtracting 7 gives 4n − 7, and "the answer is" becomes the equals sign. Solving gives 4n = 32, so n = 8.

5

HintCall the width w, write the length in terms of w, then add up all four sides.

The whyWith width w the length is w + 3, so the perimeter is 2w + 2(w + 3) = 4w + 6. Solving 4w + 6 = 26 gives w = 5 and a length of 8. Check: 5 + 8 + 5 + 8 = 26.

x + 2x + (x + 4) = 32, which simplifies to 4x + 4 = 32

HintWrite each person's payment using the same letter, then add the three payments.

The whyAli pays x, Ben 2x and Cal x + 4, and together they pay the fare. Solving 4x + 4 = 32 gives x = 7: Ali £7, Ben £14 and Cal £11, which add to £32.

The journey was 5 miles long

HintLook at which quantity in the story is being multiplied by 3.

The whyA solution is only a number until you say what it counts. Here d is the number of miles, so d = 5 answers "how far?", not "how much?". Finish every word problem with a sentence in context.

23

HintCall the smallest n and write the other two in terms of it.

The whyThe numbers are n, n + 1 and n + 2, so 3n + 3 = 72 and n = 23. The three numbers are 23, 24 and 25. Saying clearly what n stands for tells you which of them to give.