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
- Define the quantities Distinguish an item count from a unit price. Keep the same variable definitions in both conditions
- 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
- Keep exact money values Use exact decimal amounts or work in cents. Do not round during the solution
- 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
- 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
\[\begin{aligned}s&=\text{small-tub count}\\l&=\text{large-tub count}\end{aligned}\]
- \(\displaystyle s+l=32\)Use the total number of tubs
- \(\displaystyle 5s+9l=224\)Multiply each count by its unit price
- \(\displaystyle 5s+5l=160\)Multiply the count equation by five to match the small-tub terms
- \(\displaystyle \begin{gathered}(5s+9l)-(5s+5l)\\=224-160\end{gathered}\)Subtract the scaled count equation from the value equation
- \(\displaystyle 4l=64\)Collect the remaining terms
- \(\displaystyle l=16\)Divide both sides by four
- \(\displaystyle s+16=32\)Substitute the large-tub count into the original count equation
- \(\displaystyle s=16\)Subtract sixteen from both sides
- \(\displaystyle 16+16=32\)Check the total count
- \(\displaystyle 5(16)+9(16)=224\)Check the original takings. The sale contains sixteen tubs of each size. The different prices do not prevent equal counts
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 conditionReporting the other quantity
✕\(\displaystyle \text{Lantern price}=\$4\)✓\(\displaystyle \text{Lantern price}=\$7\)
For the party-shop orders, 2 balloons and 3 lanterns cost $29, while 5 balloons and 3 lanterns cost $41. Subtraction finds the balloon price first. A second step is needed to answer the lantern-price questionScaling 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 wellDifficulty 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 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, General Mathematics, chapter 5: Linear functions and models
In this chapter
Substitute values into a formulaConstruct a table of values from a formulaRecognise linear equationsSolve a linear equationSet up and solve a practical linear equationBuild and use a formula in two unknownsRead coordinates and points from a straight-line graphClassify and compare gradients from graphs5D Straight-line graphs and slope: Recognise a line or find points for its graph5D Straight-line graphs and slope: Find the gradient of a line through two points5E Straight-line equations: Recognise a line or find points for its graph5E Straight-line equations: Find the gradient of a line through two pointsRead the vertical-axis intercept and gradient from an equation5E Straight-line equations: Find the equation of a line from its graph or from its intercept and slope5F Equations from graphs and points: Find 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 equationsSet up a pair of simultaneous equations from a descriptionSolve a practical problem with simultaneous equationsUse a piecewise linear modelRead and construct step graphs