Algebra, Year 9: equations and substitution

45

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

为什么Squaring 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

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

为什么An 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

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

为什么(−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

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

为什么(−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

提示They come first in the order of operations.

为什么With 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

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

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

x = 7

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

为什么Multiplying 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

提示Nothing on either side may escape the multiplication, including the terms with no fraction in them.

为什么Multiplying 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

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

为什么Ten 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

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

为什么If 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

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

为什么Multiplying 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

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

为什么With 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

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

为什么Ali 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

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

为什么A 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

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

为什么The 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.