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Solve a practical problem with simultaneous equations

Start a set of 8
Step 3 of 12

The bars model a sale of tote bags and pencil cases. Move the tote-bag count until the takings match the recorded line. Check both neighbouring counts as well

Bars show tote-bag takings, pencil-case takings and their total. There are twenty-two items. Totes cost nine dollars and cases cost four dollars. Initially four totes and eighteen cases give takings of one hundred and eight dollars. A fixed line marks recorded takings of one hundred and fifty-three dollarsDollarsRecorded: 1530100200TotesCasesTotal3672108Takings
Total items
22
Pencil cases sold
18
Tote price
9 dollars
Pencil-case price
4 dollars
4

Which count matched the line, with its neighbours on opposite sides?

Year 11, VCE Units 1 & 2, Chapter 1, 3 difficulty levels, Calculator allowedBuild a worksheet

Use and check a contextual pair

This topic combines setting up a model with solving and interpreting it. A final value must fit both conditions and make sense for the quantity being counted or measured

  1. Define the quantities Distinguish an item count from a unit price. Keep the same variable definitions in both conditions
  2. Choose a short route Use substitution when one quantity is easy to isolate. Use elimination when a pair of coefficients already matches or can be matched with small multipliers
  3. Keep exact money values Use exact decimal amounts or work in cents. Do not round during the solution
  4. Check and interpret Check both original conditions. Report the requested quantity with its unit, and reject negative or fractional item counts when the context requires whole items
  5. State the assumption A two-price sales model assumes every item in a category has the same price and the recorded total has no extra charge or discount

Worked example

A maker sells small storage tubs for $5 each and large storage tubs for $9 each. A sale of 32 tubs brings in $224. Find the number of each size sold

\[\begin{gathered}s+l=32\\5s+9l=224\end{gathered}\]
  1. \(\displaystyle s=32-l\)Express the small-tub count in terms of the large-tub count
  2. \(\displaystyle 5(32-l)+9l=224\)Substitute into the money equation
  3. \(\displaystyle 160-5l+9l=224\)Expand the bracket
  4. \(\displaystyle 4l=64\)Collect terms and subtract 160.
  5. \(\displaystyle l=16\)Divide by 4.
  6. \(\displaystyle s=32-16=16\)Find the other count
  7. \(\displaystyle 16+16=32\)Check the original count
  8. \(\displaystyle 5(16)+9(16)=224\)Check the original takings. Sixteen tubs of each size were sold

Common mistakes

Checking only the item total
✕\(\displaystyle \begin{aligned}p&=10\\b&=10\\p+b&=20\end{aligned}\)✓\(\displaystyle \begin{aligned}7(10)+2(10)&=90\\90&\ne85\end{aligned}\)
For the pouch-and-badge model, the proposed pair satisfies the count condition but fails the takings condition
Reporting the other quantity
✕\(\displaystyle \text{Lantern price}=\$4\)✓\(\displaystyle \text{Lantern price}=\$7\)
For the party-shop orders, subtraction finds the balloon price first. A second step is needed to answer the lantern-price question
Scaling only the left side
✕\(\displaystyle 5s+5l=32\)✓\(\displaystyle 5s+5l=160\)
In the storage-tub example, multiplying the count equation by five requires multiplying its right side as well

Difficulty levels

introFind two unit prices from two short purchase records
standardFind the requested item count from a count total and takings
examChoose the requested count from a short practical problem without setup prompts

Questions about this topic

Must I use elimination?+

No. Use substitution, elimination or another valid method. Check the final pair against both original conditions

Why check the context after solving?+

The equations may give values that do not fit a physical situation. A count of whole items cannot be negative or fractional

What if the two prices are equal?+

For a count-and-value model, the money total then repeats the count information or contradicts it. It cannot determine the split. These practice questions use different prices

Can I round a fractional item count?+

No. If the context requires whole items, a fractional solution means the stated model or data needs checking. Rounding generally breaks one or both equations

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 1: Linear relations and coordinate geometry

In this chapter

Recognise linear equationsSubstitute values into a formulaSolve a linear equationSet up and solve a practical linear equationRearrange literal equations and construct equivalent line formsConstruct a table of values from a formulaRead coordinates and points from a straight-line graphClassify and compare gradients from graphsFind the gradient of a line through two pointsSolve and interpret a linear inequationRead the vertical-axis intercept and gradient from an equationRecognise a line or find points for its graphConstruct a line on a coordinate gridFind the equation of a line from its graph or from its intercept and slopeFind the equation of a line through two pointsConstruct a linear model from a descriptionInterpret the graph of a linear modelRead the point of intersection of two linesSolve a pair of simultaneous linear equationsSolve a pair of simultaneous linear equations by eliminationSet up a pair of simultaneous equations from a descriptionSolve a practical problem with simultaneous equationsSolve a linear system with three or four unknownsVerify coordinate incidence and determine a parameterForm an equation of a parallel or perpendicular lineRelate a line's gradient and angle of inclinationMidpoint, endpoint and distance of a coordinate segmentDetermine a perpendicular bisector