Rearrange literal equations and construct equivalent line forms

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?

Starting equation

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

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

Consider P = an + b with a = 0 and P = b. What can be concluded about n?

Consider P = an + b with a = 0 and P ≠ b. What can be concluded about n?

Equation and condition

\(\displaystyle ax+b=cx+d,\quad a\ne c\)

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

Read the coefficient and the constant

Make the vertical variable the subject before reading the gradient and intercept. Either order of terms is valid

1. 1Gradient Keep the sign of the horizontal-variable coefficient. It gives the vertical change for one unit to the right \(\displaystyle y=11-4x\quad\Rightarrow\quad m=-4\)
2. 2Vertical-axis intercept Set the horizontal coordinate to zero. The constant is the vertical coordinate, and the crossing is a point \(\displaystyle \begin{aligned}x&=0\\y&=c\\P&=(0,c)\end{aligned}\)
3. 3Missing terms No constant means a zero intercept. No horizontal-variable term means a zero gradient \(\displaystyle \begin{aligned}y&=-6x+0\\y&=0x+9\end{aligned}\)
4. 4Divide every term First remove the horizontal-variable term from the left. Then divide every term by the vertical-variable coefficient \(\displaystyle \begin{aligned}5y+7x&=30\\5y&=30-7x\\y&=6-\frac{7}{5}x\end{aligned}\)

One line, different forms

A line can have many equivalent equations. General form places all terms on one side. To specify one coefficient triple, clear fractions, remove the common integer factor and make the first non-zero coefficient positive

1. 1Move the complete expression Subtract the whole right side, keeping every sign \(\displaystyle \begin{aligned}y&=-\frac32x+4\\\frac32x+y-4&=0\end{aligned}\)
2. 2Clear denominators Multiply every term by 2. This gives integer coefficients with no common factor greater than 1. \(\displaystyle 3x+2y-8=0\)
3. 3Equivalent, but not canonical Doubling every coefficient keeps the same line, but the resulting triple has a common factor of 2. \(\displaystyle 6x+4y-16=0\)

Convert to Ax + By + C = 0 with primitive integer coefficients and the first non-zero coefficient positive. Find C.

A point and a gradient

For a non-vertical line through the stated point, use the point-gradient rule shown above. Substitute the signed coordinates before expanding. This changes the equation's form; graph construction is a later task

1. 1Keep a negative coordinate Subtracting negative two gives an addition. The vertical coordinate is five and the gradient is three halves \(\displaystyle y-5=\frac32(x+2)\)
2. 2Expand, then isolate Multiply both terms inside the bracket before adding 5 to both sides \(\displaystyle \begin{aligned}y-5&=\frac32x+3\\y&=\frac32x+8\end{aligned}\)
3. 3Check the original point At x = −2, the new rule gives y = 5, as required \(\displaystyle \frac32(-2)+8=5\)

Rewrite in gradient-intercept form y = mx + c. Find the constant c.

Which description matches this equation?

Which equation can be written in gradient-intercept form with a finite gradient?

Horizontal and vertical equations

A horizontal line has gradient zero. A vertical line has a fixed horizontal coordinate and no finite gradient, so it cannot be written in gradient-intercept form. General form handles both

1. 1Horizontal There is no x term; its coefficient is zero \(\displaystyle y=-4\quad\Longleftrightarrow\quad0x+y+4=0\)
2. 2Vertical There is no y term; its coefficient is zero \(\displaystyle x=6\quad\Longleftrightarrow\quad x+0y-6=0\)
3. 3Whole equation entry For a new subject, enter a complete equality. Equivalent expanded forms are valid. Current practice has numerical coefficients and does not ask you to divide by a variable expression \(\displaystyle q=\frac{2r-12}{3}\)