Substitute into and rearrange a formula

Formula

\(\displaystyle y=3(x+6)\)

Known output

\(\displaystyle y=45\)

Find the value of the unknown

Original formula

\(\displaystyle y=a(x+6)\)

Given output

\(\displaystyle y=12\)

First attempt

\(\displaystyle x=\frac{12}{a}-6\)

Second attempt

\(\displaystyle x=\frac{12-6}{a}\)

Each attempt gives a proposed input. Substitute it into the original formula to rebuild the output. Increase the multiplier and compare both rebuilt outputs with the given output

As the multiplier increases, which attempt continues to rebuild the given output?

A value or a new subject

Finding a value gives one numerical answer. Making a new subject keeps the other variables, so the result describes the whole relationship. Use the equation-solving steps you already know on both sides

1. 1Recover the bracket Divide both sides by the outside coefficient \(\displaystyle \frac{y}{3}=x+6\)
2. 2Isolate the target Subtract the offset from both sides \(\displaystyle x=\frac{y}{3}-6\)
3. 3Other valid routes Expanding first also works. Subtract the full expanded constant before dividing \(\displaystyle \begin{aligned}y&=3x+18\\y-18&=3x\\x&=\frac{y-18}{3}\end{aligned}\)

Formula

\(\displaystyle y=-4(x-2)\)

Known output

\(\displaystyle y=28\)

Find the value of the unknown

First recover the value of the bracket. Keep the negative divisor

Formula

\(\displaystyle P=5q-7\)

Target

\(\displaystyle q\)

Which formula makes the target the subject?

Which line contains the first error?

Targets in a fraction

Clear a denominator by multiplying both sides. The next operation depends on the target. A denominator cannot be zero, and a new division needs a nonzero divisor

1. 1Clear the fraction Multiply both sides by the denominator \(\displaystyle ry=p-q\)
2. 2Target in the numerator Add the target to both sides, then subtract the product \(\displaystyle \begin{aligned}ry+q&=p\\q&=p-ry\end{aligned}\)
3. 3Target in the denominator If the output is nonzero, divide by it to isolate the denominator variable \(\displaystyle r=\frac{p-q}{y},\quad y\ne0\)

Formula

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

Conditions

\(\displaystyle u\ne0,\quad v\ne0\)

Target

\(\displaystyle u\)

Which formula makes the target the subject?

A non-negative squared length

Square both sides to remove the square root. Isolate the squared target before taking a root. The length condition selects the non-negative root

1. 1Square both sides The root contains the whole sum \(\displaystyle d^2=a^2+b^2\)
2. 2Isolate the square Subtract the other squared length from both sides \(\displaystyle a^2=d^2-b^2\)
3. 3Use the length condition Take the non-negative root of the whole difference \(\displaystyle a=\sqrt{d^2-b^2}\)

Length relationship

\(\displaystyle d=\sqrt{a^2+b^2}\)

Known lengths

\(\displaystyle d=25\text{ cm},\quad b=7\text{ cm}\)

Target condition

\(\displaystyle a\ge0\)

Find the non-negative missing length

Square both known lengths before subtracting. Then take the non-negative root

Length conditions

\(\displaystyle r\ge0,\quad s\ge0,\quad h\ge s\)

Which formula corrects the student's final line?

Equivalent routes and exceptional cases

Inverse operations give a convenient route, not the only valid route. Apply each operation to the whole of both sides. Before dividing by an expression containing variables, separate the case when that expression is zero

1. 1Divide all terms Dividing first is valid if the constant is also divided \(\displaystyle \begin{aligned}P&=4n+7\\\frac{P}{4}&=n+\frac74\end{aligned}\)
2. 2Factor a repeated target Collect the target terms and factor before division \(\displaystyle \begin{aligned}ax+b&=cx+d\\(a-c)x&=d-b\end{aligned}\)
3. 3State the condition The quotient requires a nonzero coefficient. If it is zero, inspect the remaining equation \(\displaystyle x=\frac{d-b}{a-c},\quad a\ne c\)
4. 4A square can have two roots A non-negative length condition selects one branch. Without that condition, retain both signs when the radicand is positive \(\displaystyle x^2=36\Rightarrow x=\pm6\)

Starting formula

\(\displaystyle P=4n+7\)

A student divides both complete sides of the formula by 4. Is this step valid?

Formula and values

\(\displaystyle y=x^2+5,\quad y=41,\ x\ge0\)

The unknown represents a non-negative length. Find it from the squared formula

Formula and output

\(\displaystyle y=x^2+5,\quad y=41\)

Now the unknown is an unrestricted real number. Which complete answer solves the formula for the given output?

Formula and conditions

\(\displaystyle P=an+b,\quad a=0,\quad P=b\)

Consider the formula and conditions below. What can be concluded about the unknown?

Formula and conditions

\(\displaystyle P=an+b,\quad a=0,\quad P\ne b\)

Consider the formula and conditions below. What can be concluded about the unknown?

For ax + b = cx + d with a ≠ c, which formula makes x the subject?

Formula and values

\(\displaystyle A=\frac{(a+b)h}{2}\)

A trapezium has parallel sides 7 cm and 11 cm and area 72 square centimetres. Rearrange the area formula to find the perpendicular height