Year 7 Mathematics · AC9M7A01

Variables, Formulas and Substitution

We are learning to recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown

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Learning goalsSay it simply

We are learning to recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown.

A variable is a symbol representing a quantity that may change or be unknown. A formula states a general relationship among quantities, such as taxi cost C = 4.50 + 2.20d.

The fixed term and variable term have different roles: $4.50 is charged once, while $2.20 is multiplied by distance d. Units help explain each symbol and coefficient.

Substitution evaluates the formula for a particular case. Replace each variable with its given value, retain grouping, apply operation order and report the resulting quantity with its unit.

Success criteria

  • I can define variables and represent an everyday relationship with a formula.
  • I can explain the fixed and variable terms in a formula.
  • I can substitute values, calculate accurately and report the correct units.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Substitute into a taxi formula

  1. Use C=4.50+2.20d with d=8 km.
  2. Substitute: C=4.50+2.20×8.
  3. Multiply first: C=4.50+17.60.
  4. Add to obtain C=$22.10.

Final answer: The taxi cost is $22.10.

Check: Substitute d=8 back into the original formula.

Example 2

Represent and use a perimeter formula

  1. A rectangle has length l=8 cm and width w=5 cm.
  2. Use P=2l+2w.
  3. Substitute P=2×8+2×5.
  4. Calculate P=16+10=26 cm.

Final answer: The perimeter is 26 cm.

Check: Adding 8+5+8+5 also gives 26 cm.

Example 3

Application problem 1

Problem: A taxi fare is C = 4.50 + 2.20d, where d is kilometres travelled. Find the cost of a 13 km trip and explain what each number in the formula represents.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: C = 4.50 + 2.20×13 = $33.10. The 4.50 is the fixed charge and 2.20 is the cost per kilometre.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: C = 4.50 + 2.20×13 = $33.10. The 4.50 is the fixed charge and 2.20 is the cost per kilometre.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: Create a formula for a $40 setup fee plus $12 per item, then find the cost of 15 items.

  1. Plan: Define the variable, write the formula, then substitute 15.
  2. Work: Let n be the number of items. Then C=40+12n, and for 15 items C=40+12×15=$220.
  3. Interpret: A formula represents the relationship before evaluating a case.

Final answer: Let n be the number of items. Then C=40+12n, and for 15 items C=40+12×15=$220.

Check: A formula represents the relationship before evaluating a case.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7A01 - Variables, Formulas and Substitution
Example 1

Example 1 Substitute into a taxi formula Use C=4.50+2.20d with d=8 km. Substitute: C=4.50+2.20×8. Multiply first: C=4.50+17.60. Add to obtain C=$22.10. Final answer: The taxi cost is $22.10. Check: Substitute d=8 back into the original formula.

Example 2

Example 2 Represent and use a perimeter formula A rectangle has length l=8 cm and width w=5 cm. Use P=2l+2w. Substitute P=2×8+2×5. Calculate P=16+10=26 cm. Final answer: The perimeter is 26 cm. Check: Adding 8+5+8+5 also gives 26 cm.

Example 3

Example 3 Application problem 1 Problem: A taxi fare is C = 4.50 + 2.20d, where d is kilometres travelled. Find the cost of a 13 km trip and explain what each number in the formula represents. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: C = 4.50 + 2.20×13 = $33.10. The 4.50 is the fixed charge and 2.20 is the cost per kilometre. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: C = 4.50 + 2.20×13 = $33.10. The 4.50 is the fixed charge and 2.20 is the cost per kilometre. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: Create a formula for a $40 setup fee plus $12 per item, then find the cost of 15 items. Plan: Define the variable, write the formula, then substitute 15. Work: Let n be the number of items. Then C=40+12n, and for 15 items C=40+12×15=$220. Interpret: A formula represents the relationship before evaluating a case. Final answer: Let n be the number of items. Then C=40+12n, and for 15 items C=40+12×15=$220. Check: A formula represents the relationship before evaluating a case.

Curriculum examplesCopied content

The content description and elaborations below show the curriculum ideas taught in this unit. Items marked as teaching context support lesson planning.

  • Content description: recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown
  • E1: linking variables to attributes and measures being modelled when using formulas, such as the area of a rectangle is equal to the length x width as A\;=\;l\;\times\;w or using p\;=\;6g\;+\;b to describe a total of points expressed as goals (worth 6 points) and behinds (worth one point)
  • E2: interpreting and using formulas obtained from other sources; for example, maximum heart rates and target heart rates for moderate exercise
  • E3: substituting numerical values for variables when using formulas and calculating the value of an unknown in practical situations; for example, calculating weekly wage W given base wage b and overtime hours h at 1.5 times rate r, W\;=\;b+1.5\times h\times r, using values for mass m and volume v to determine density d of a substance where d\;=\;\frac mv
  • E4: using everyday formulas and their application to contexts on Country/Place, investigating the relationships between variables
Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. In C=4.50+2.20d, identify the variable and its meaning.

    Check answer

    Answer: d is the variable and represents distance travelled in kilometres.

    Hint: Use the context and units.

    Why: A variable represents a quantity that can change.

  2. 2. Use P=2l+2w when l=8 cm and w=5 cm.

    Check answer

    Answer: P=2×8+2×5=26 cm.

    Hint: Use brackets or multiplication signs clearly.

    Why: Substitution replaces each variable with its value.

  3. 3. Use A=lw when l=12 m and w=7 m.

    Check answer

    Answer: A=12×7=84 m².

    Hint: Substitute both dimensions before multiplying.

    Why: The formula represents rectangle area.

  4. 4. A taxi costs C=4.50+2.20d. Find C for d=8 km.

    Check answer

    Answer: C=4.50+2.20×8=$22.10.

    Hint: Multiply the rate by distance, then add the fixed charge.

    Why: The fixed charge and distance charge are both included.

  5. 5. The formula T=3n+5 gives points for n completed tasks. Find T when n=9.

    Check answer

    Answer: T=3×9+5=32 points.

    Hint: Replace n with 9 and follow operation order.

    Why: Substitution evaluates one case of a general formula.

  6. 6. Write a formula for total cost C of x tickets at $7 each plus a $3 booking fee.

    Check answer

    Answer: C=7x+3.

    Hint: Multiply ticket count by 7, then add 3.

    Why: The variable charge and fixed fee form different terms.

  7. 7. A rectangle has area A=lw. Find A when l=6.5 m and w=4 m.

    Check answer

    Answer: A=6.5×4=26 m².

    Hint: Write the substituted formula before calculating.

    Why: Area retains square units after substitution.

  8. 8. A hire cost is H=25+6h. Compare 3 hours and 7 hours.

    Check answer

    Answer: H(3)=25+18=$43; H(7)=25+42=$67; the difference is $24.

    Hint: Evaluate the formula twice, then subtract.

    Why: The fixed fee is unchanged while the hourly term changes.

  9. 9. Create a formula for a $40 setup fee plus $12 per item, then find the cost of 15 items.

    Check answer

    Answer: Let n be the number of items. Then C=40+12n, and for 15 items C=40+12×15=$220.

    Hint: Define the variable, write the formula, then substitute 15.

    Why: A formula represents the relationship before evaluating a case.

  10. 10. A taxi fare is C = 4.50 + 2.20d, where d is kilometres travelled. Find the cost of a 13 km trip and explain what each number in the formula represents.

    Check answer

    Answer: C = 4.50 + 2.20×13 = $33.10. The 4.50 is the fixed charge and 2.20 is the cost per kilometre.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: Variable treated as a label only.

Correction: It represents a numerical quantity with possible units.

Common mistake: Adjacent symbols added.

Correction: In algebra 2d means 2×d.

Common mistake: Units mixed during substitution.

Correction: Convert quantities first.

Curriculum alignmentStart here

We are learning to recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown.

A variable is a symbol representing a quantity that may change or be unknown. A formula states a general relationship among quantities, such as taxi cost C = 4.50 + 2.20d.

The fixed term and variable term have different roles: $4.50 is charged once, while $2.20 is multiplied by distance d. Units help explain each symbol and coefficient.

Substitution evaluates the formula for a particular case. Replace each variable with its given value, retain grouping, apply operation order and report the resulting quantity with its unit.

Success criteria

  • I can define variables and represent an everyday relationship with a formula.
  • I can explain the fixed and variable terms in a formula.
  • I can substitute values, calculate accurately and report the correct units.
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