Rearrange literal equations and recognise equivalent line forms

Perimeter rule

\(\displaystyle P=2l+2w\)

A rectangular yard is fenced. Its perimeter follows the rule shown, where l is the length and w is the width. The fencing job needs the width in terms of the perimeter P and the length l. Which is the rearranged rule?

Move the multiplier from 1 up to 5 and compare the last two columns with the original column. Watch the readouts for y at x = 0 as the multiplier changes

Which multiplied equation matched the original at every x for every position of the multiplier?

Equivalent linear forms

Two equations are equivalent when they have the same solutions. Multiplying or dividing every term by the same non-zero number keeps the line, while multiplying the x and y terms but not the constant term gives a different line

1. 1Apply to both sides Add, subtract, multiply or divide on both sides of the equality \(\displaystyle P=2l+2w\Rightarrow P-2l=2w\)
2. 2Move non-target terms Use inverse addition or subtraction to collect the target term on one side \(\displaystyle P=2l+2w\Rightarrow P-2l=2w\)
3. 3Divide the whole coefficient Divide every term by the coefficient of the target symbol, then simplify each term \(\displaystyle w=\frac{P-2l}{2}=\frac{P}{2}-l\)
4. 4Do not cancel across a sum A symbol added to another symbol cannot be dropped. Subtract it from the other side instead \(\displaystyle l+w=12\Rightarrow w=12-l\)

Speed rule

\(\displaystyle v=u+at\)

Start speed

\(\displaystyle 8\text{ m/s}\)

Acceleration

\(\displaystyle 3\text{ m/s}^2\)

Target speed

\(\displaystyle 26\text{ m/s}\)

A delivery van starts at 8 m/s and accelerates at 3 m/s squared towards 26 m/s. Rearrange the speed rule for t and find the time to reach 26 m/s.

Subtract u from both sides, then divide every term by a.

Distance rule

\(\displaystyle s=\frac{(u+v)t}{2}\)

Speed at start of braking

\(\displaystyle 26\text{ m/s}\)

Speed at end of braking

\(\displaystyle 11\text{ m/s}\)

Braking distance

\(\displaystyle 74\text{ m}\)

The same van then brakes uniformly from 26 m/s to 11 m/s over 74 m. Rearrange the distance rule for t and find the braking time

Multiply both sides by 2, then divide by (u + v). Keep the bracket together

Which line correctly subtracts 5 from both sides?