Algebra, Year 10: simultaneous equations and iteration

x = 7 and y = 3

提示One equation has +y and the other has −y. Think what happens if the two equations are combined.

为什么Adding the equations makes the y terms cancel: 2x = 14, so x = 7. Putting x = 7 into x + y = 10 gives y = 3. Check in the other equation: 7 − 3 = 4.

12; 4; 2

提示Doubling makes the y terms match, so that taking one equation from the other removes them.

为什么The y terms must match before they can be eliminated. After doubling, (3x + 2y) − (2x + 2y) = x and 16 − 12 = 4. Then x + y = 6 gives y = 2. Check in the first equation: 3 × 4 + 2 × 2 = 16.

x = 5 and y = 15

提示The first equation already says what y is worth in terms of x. Use that in the second equation.

为什么Replacing y by 3x in the second equation gives x + 3x = 20, so 4x = 20 and x = 5. Then y = 3 × 5 = 15. This method is called substitution, and it is quickest when one equation already has a letter as its subject.

A coffee costs £2.50 and a tea costs £2.00

提示Compare the two orders: what is in the first one that is not in the second?

为什么With c for a coffee and t for a tea: 2c + t = 7 and c + t = 4.5. Subtracting the second from the first leaves one coffee: c = 2.50. Then 2.50 + t = 4.50, so t = 2.00. Check: 2 × 2.50 + 2.00 = 7.00.

Elimination

提示The word means getting rid of something, as when a team is knocked out of a cup.

为什么If the two equations have the same number of x or of y, adding or subtracting them leaves an equation with only one unknown. If they do not match, multiply one or both equations first so that they do.

Iteration

提示The word comes from the Latin for "again".

为什么Some equations cannot be solved exactly by rearranging. Instead a formula such as xₙ₊₁ = (xₙ + 6)/2 is applied to a starting value, then to the result, and so on, until the values settle down to the solution.

4; 5; 5.5

提示Each new value is made from the one just before it, not from the starting value.

为什么x₁ = (2 + 6)/2, x₂ = (4 + 6)/2 and x₃ = (5 + 6)/2. The small numbers are labels that count the steps; they are not powers or multipliers. The values are closing in on 6, the solution of x = (x + 6)/2.

179

提示Work out the first year, then use that result as the start of the second year.

为什么P₁ = 1.1 × 200 − 30 = 190. Then P₂ = 1.1 × 190 − 30 = 209 − 30 = 179. The same rule is applied again to each new answer, which is what makes it an iterative process.

It has a solution between x = 1 and x = 2

提示One output is negative and the other positive. Think about what the graph must do on the way from one to the other.

为什么The graph of y = x³ + x − 5 is below the x-axis at x = 1 and above it at x = 2. It is a continuous curve, so it must cross the axis somewhere in between, and a crossing is a solution. Iteration can then pin the value down (it is about 1.52).

When two values in a row agree once they are rounded to 2 decimal places

提示Watch how much each new value differs from the one before.

为什么As an iteration closes in on a solution, the values change less and less. Once the change is too small to alter the second decimal place, further steps will not change the rounded answer. Keep all the calculator digits while iterating and round only at the end.

Add 3x + 1 to both sides to get x³ = 3x + 1, then take the cube root of both sides

提示Aim to leave only the cubed term on the left.

为什么x³ − 3x − 1 = 0 becomes x³ = 3x + 1, and taking the cube root gives x = ∛(3x + 1). An x on each side is what an iteration needs: the formula becomes xₙ₊₁ = ∛(3xₙ + 1), with the old value on the right and the new one on the left.