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Set up a pair of simultaneous equations from a description

Start a set of 8
Step 3 of 12

Give each condition its own equation

First name the unknown quantities. Keep each letter's meaning fixed. Decide what each condition counts or measures before writing the pair

  1. 1Two purchases When the unknowns are unit prices, each purchase has its own item counts and total. Here the letters name the two unit prices and the coefficients name the purchased counts \(\displaystyle \begin{aligned}ax+by&=P\\cx+dy&=Q\end{aligned}\)
  2. 2Count and value When the unknowns are item counts, the first equation adds the counts. The second weights them by their unit prices. Here the capital letters name the total count and total value \(\displaystyle \begin{aligned}x+y&=N\\px+qy&=V\end{aligned}\)
  3. 3A stated difference Subtract the smaller quantity from the larger. Here the first quantity is greater by the stated difference \(\displaystyle x-y=D\)
  4. 4A future age total Both people age by the elapsed time. Here the letters name current ages, elapsed years and the future total \(\displaystyle (x+t)+(y+t)=S\)
Year 10, Chapter 8, 4 difficulty levels, No calculatorBuild a worksheet

Represent both conditions

A simultaneous pair describes the same unknowns under two conditions. Define each unknown and its unit before translating the conditions. This topic stops at the equations

  1. Keep meanings fixed Either letter can name either quantity. What matters is using your definitions consistently in both equations
  2. Check the units Counts add to a count. A unit price multiplied by an item count gives a value in dollars
  3. Use different information Doubling an equation repeats its condition. It does not replace the other condition in the description
  4. Translate change for each quantity A future age total includes the elapsed years for both people. The difference between their ages stays the same \(\displaystyle \begin{aligned}(x+t)+(y+t)&=S\\x+y+2t&=S\\x+y&=S-2t\end{aligned}\)

Worked example

A library shop sells greeting cards and reading journals. An order of six cards and two journals costs $40. Another order of three cards and five journals costs $64. Form a pair of equations for the two unit prices

\[\begin{aligned}c&=\text{card price in dollars}\\j&=\text{journal price in dollars}\end{aligned}\]
  1. \(\displaystyle 6c+2j=40\)Multiply each unit price by its count in the first order
  2. \(\displaystyle 3c+5j=64\)Use the same price definitions for the second order

Common mistakes

Swapping quantities
✕\(\displaystyle 2c+6j=40\)✓\(\displaystyle 6c+2j=40\)
For the first library-shop order, there are six cards and two journals. Keep each count with its own unit price
Using a count as a value total
✕\(\displaystyle 3s+7l=29\)✓\(\displaystyle s+l=29\)
Here the letters count small and large pennants and the stated total counts pennants. Prices belong in the separate money equation
Advancing only one age
✕\(\displaystyle (x+t)+y=S\)✓\(\displaystyle (x+t)+(y+t)=S\)
The letters represent two current ages. Each age must increase by the elapsed time before adding them

Difficulty levels

introComplete equations for two purchases with fixed variable meanings
standardComplete count-and-value or sum-and-difference equations
challengeAccount for both people's changing ages when completing or recognising a pair
examRecognise a pair for two purchases without a completion scaffold

Questions about this topic

Can I choose different letters?+

Yes. State what each letter represents and keep that meaning throughout the pair

Does every pair use a count equation?+

No. Two purchases give two value equations. Other descriptions may give a total and a difference, or a current relation and a future total

Do I need to solve the pair here?+

No. This topic practises representing the conditions. Practical simultaneous problems use the equations to find the requested quantities

Do two written equations always determine two unknowns?+

No. One may repeat the other, or they may contradict. For two straight lines, a unique pair requires nonparallel lines

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 8: Linear equations and inequalities

In this chapter

8A Linear equations and expressions: Solve a linear equationRecognise linear equations8A Linear equations and expressions: Set up and solve a practical linear equation8B Equations with brackets and fractions: Solve a linear equation8B Equations with brackets and fractions: Set up and solve a practical linear equationSolve equations that contain algebraic fractionsSubstitute values into a formulaSubstitute into and rearrange a formulaRead, draw and match inequality number lines8D Linear inequalities: Solve and interpret a linear inequationRead the point of intersection of two linesSolve a pair of simultaneous linear equationsSolve simultaneous equations by substitution8F Simultaneous equations by substitution: Solve a practical problem with simultaneous equations8F Simultaneous equations by substitution: Set up a pair of simultaneous equations from a description8F Simultaneous equations by substitution: Build and use a formula in two unknownsSolve simultaneous equations by elimination8G Simultaneous equations by elimination: Solve a practical problem with simultaneous equations8G Simultaneous equations by elimination: Set up a pair of simultaneous equations from a description8G Simultaneous equations by elimination: Build and use a formula in two unknowns8H Linear inequalities in the coordinate plane: Solve and interpret a linear inequation