Search topics

Search

Rearrange literal equations and recognise equivalent line forms

Start a set of 8
Step 3 of 6

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

\[\text{same operation on every term}\]
  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 \(\displaystyle l+w=12\Rightarrow w=12-l\)
Year 12, VCE Units 3 & 4, Chapter 14, 4 difficulty levels, No calculatorBuild a worksheet

Rearrange an equation by equivalent operations

Equivalent equations have the same solutions. Keep both sides balanced and simplify only after applying an operation to the whole equation

\[\text{same operation on both sides}\]
  1. Apply to both sides Add, subtract, multiply or divide on both sides of the equality \(\displaystyle a=b\Rightarrow a+3=b+3\)
  2. Move non-target terms Use inverse addition or subtraction to collect the target term \(\displaystyle 2p+6=q+2\Rightarrow2p=q-4\)
  3. Divide the whole coefficient Divide every term by the coefficient of the target symbol \(\displaystyle 2p=q-4\Rightarrow p=\frac{1}{2}q-2\)
  4. Do not cancel across a sum Cancellation is valid for common factors, not for separate addends \(\displaystyle p+q=8\Rightarrow p=8-q\)

Worked example

Rearrange the literal equation to make q the subject

\[3q+8=2r-4\]
  1. \(\displaystyle 3q+8=2r-4\)Start with the stated equation
  2. \(\displaystyle 3q=2r-12\)Subtract 8 from both sides
  3. \(\displaystyle q=\frac{2r-12}{3}\)Divide every term by 3.
  4. \(\displaystyle q=\frac{2}{3}r-4\)Simplify each term without changing the equation

Common mistakes

Applying an operation to one side
✕\(\displaystyle 2p+6=q+2\Rightarrow2p=q+2\)✓\(\displaystyle 2p+6=q+2\Rightarrow2p=q-4\)
The same operation must be applied to both sides of an equation
Cancelling across a sum
✕\(\displaystyle p+q=8\Rightarrow q=8\)✓\(\displaystyle p+q=8\Rightarrow q=8-p\)
A symbol in a sum is not a common factor that can be cancelled
Scaling only the x and y terms of a line equation
✕\(\displaystyle 2x-y+3=0\Rightarrow4x-2y+3=0\)✓\(\displaystyle 2x-y+3=0\Rightarrow4x-2y+6=0\)
Multiplying an equation by 2 requires every term to be multiplied by 2.

Difficulty levels

introConvert a numerical line equation between equivalent line forms
standardIsolate a symbol in a literal equation with one denominator or a bracket structure
challengeRearrange several literal terms or handle horizontal and vertical line forms
examSelect the equivalent formula after one named algebraic error has been introduced into each distractor

Questions about this topic

Why must the same operation be applied to both sides?+

An equation states that its two sides have equal values. Applying the same operation to both sides preserves that equality

When can a factor be cancelled?+

A factor can be cancelled from a product or quotient when it is a common non-zero factor. Separate terms in a sum cannot be cancelled

How is a vertical line represented?+

A vertical line has a constant x-coordinate, such as an x-coordinate of 4. It does not have a finite gradient-intercept form

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 14: Statistical inference

In this chapter

Populations, samples, parameters, statistics and sampling methodsAllocate a stratified random sampleFind a percentage of a quantityCalculate a sample proportion and list its possible valuesExact sampling distribution of a sample proportionFinite events, probability assignments and complementsConditional probability from exact representationsEvaluate factorial and combination expressionsConvert between fractions, decimals and percentagesLarge-sample conditions and the sample-proportion distributionCalculate a standard score or actual valueUse z-score boundaries to estimate normal-model percentages and countsCalculate the standard deviation from the formulaSolve a linear equationSolve a quadratic exactly with the quadratic formulaApproximate confidence intervals for a population proportionRearrange literal equations and recognise equivalent line formsInterpret simulated sampling distributions and interval variationDesign and interpret a probability simulation