Simultaneous linear equations
Solve pairs of linear equations graphically and algebraically; identify the intersection point as a break-even point; model and interpret practical problems involving cost versus revenue and payment plans.
Worked examples
Solving algebraically (elimination)
Straightforward
Problem
Solve the simultaneous equations and . Find the values of and .
1
Look for a variable to eliminate. Both equations have (with opposite signs), so adding them eliminates .
2
Simplify and solve for .
3
Substitute into the simpler equation and solve for .
4
Check the solution in both original equations.
Check: ✓ and ✓
Answer
The solution is , . The lines intersect at the point .
Break-even analysis
Moderate
Problem
A food truck has weekly fixed costs of $300 and variable costs of $4 per meal. Each meal is sold for $9. How many meals must be sold each week to break even?
1
Write the revenue and cost equations in terms of , the number of meals sold.
2
The break-even point is where revenue equals cost. Set .
3
Solve for .
4
Interpret the result.
The food truck breaks even at 60 meals per week. Selling more than 60 meals generates a profit; fewer than 60 results in a loss.
Answer
The food truck breaks even at 60 meals sold per week.
Comparing two options
Challenging
Problem
Two internet providers offer the following deals. Provider A: $25 per month plus $0.50 per GB of data. Provider B: $10 per month plus $1.00 per GB. After how many GB do the plans cost the same, and which plan is cheaper for heavy data users?
1
Write a cost equation for each provider in terms of (gigabytes used per month).
2
Set the two costs equal to find the break-even usage.
3
Solve for .
4
Interpret the result for usage above and below the break-even point.
At 30 GB the plans cost the same (both cost $40). Below 30 GB, Provider B is cheaper. Above 30 GB, Provider A is cheaper — so Provider A is better for heavy users.
Answer
The plans cost the same at 30 GB ($40). Provider A is cheaper for heavy data users (above 30 GB).
Practise
Q1·Straightforward
Solve the simultaneous equations and . Find the value of .
Explanation
Adding the equations:
Q2·Straightforward
Solve the simultaneous equations and . Find the value of .
Explanation
Subtracting the second equation from the first:
Q3·Straightforward
A company's revenue is dollars and its total cost is dollars, where is the number of units sold. Find the break-even quantity (the value of where ).
Explanation
Set revenue equal to cost:
The company breaks even at 5 units.
The company breaks even at 5 units.
Q4·Moderate
Phone Plan A charges $0.40 per minute plus $15 per month. Phone Plan B charges $0.80 per minute plus $5 per month. After how many minutes of calls per month do the two plans cost the same amount?
Explanation
Let be the number of minutes. Set the costs equal:
The plans cost the same at 25 minutes per month.
The plans cost the same at 25 minutes per month.
Q5·Moderate
Solve the simultaneous equations and . Find the value of .
Explanation
Adding the two equations:
Q6·Moderate
Two numbers have a sum of 50 and a difference of 8. Find the larger of the two numbers.
Explanation
Let the larger number be and the smaller be :
Adding:
The larger number is 29.
Adding:
The larger number is 29.
Q7·Moderate
A business has revenue dollars and total cost dollars, where is the number of items produced. How many items must be produced to break even?
Explanation
Set :
The business breaks even at 12 items.
The business breaks even at 12 items.
Q8·Moderate
Two lines intersect at a point. The lines have equations and . Find the -value at the point of intersection.
Explanation
Set the equations equal:
Substitute into :
The lines intersect at .
Substitute into :
The lines intersect at .
Q9·Challenging
A school canteen sells drinks at $3 each and snacks at $5 each. On one day they sell 60 items in total and take $236. How many drinks were sold?
Explanation
Let = drinks sold and = snacks sold.
Total items:
Total revenue:
From the first equation:
Substitute into the revenue equation:
32 drinks were sold.
Total items:
Total revenue:
From the first equation:
Substitute into the revenue equation:
32 drinks were sold.
Q10·Challenging
A plumber charges a $200 call-out fee plus $15 per hour. An electrician charges no call-out fee but $40 per hour. After how many hours of work do they charge the same total amount?
Explanation
Let = hours of work.
Plumber:
Electrician:
Set equal:
After 8 hours both charge the same amount.
Plumber:
Electrician:
Set equal:
After 8 hours both charge the same amount.
Q11·Challenging
Solve the simultaneous equations and . Find the value of .
Explanation
From the second equation:
Substitute into the first:
Then .
So .
Substitute into the first:
Then .
So .
Q12·Challenging
A plumber charges $80 call-out plus $60 per hour. An electrician charges $50 call-out plus $75 per hour. After how many hours does the electrician become more expensive than the plumber? Give the number of hours at which they cost the same.
Explanation
Let = hours of work.
Plumber:
Electrician:
Set equal:
At 2 hours the costs are equal. Beyond 2 hours the electrician charges more.
Plumber:
Electrician:
Set equal:
At 2 hours the costs are equal. Beyond 2 hours the electrician charges more.
Open Math
Simultaneous linear equations
Algebra · MS-A4
Name:
Date:
Q1Straightforward
Solve the simultaneous equations and . Find the value of .
Q2Straightforward
Solve the simultaneous equations and . Find the value of .
Q3Straightforward
A company's revenue is dollars and its total cost is dollars, where is the number of units sold. Find the break-even quantity (the value of where ).
Q4Moderate
Phone Plan A charges $0.40 per minute plus $15 per month. Phone Plan B charges $0.80 per minute plus $5 per month. After how many minutes of calls per month do the two plans cost the same amount?
Q5Moderate
Solve the simultaneous equations and . Find the value of .
Q6Moderate
Two numbers have a sum of 50 and a difference of 8. Find the larger of the two numbers.
Q7Moderate
A business has revenue dollars and total cost dollars, where is the number of items produced. How many items must be produced to break even?
Q8Moderate
Two lines intersect at a point. The lines have equations and . Find the -value at the point of intersection.
Q9Challenging
A school canteen sells drinks at $3 each and snacks at $5 each. On one day they sell 60 items in total and take $236. How many drinks were sold?
Q10Challenging
A plumber charges a $200 call-out fee plus $15 per hour. An electrician charges no call-out fee but $40 per hour. After how many hours of work do they charge the same total amount?
Q11Challenging
Solve the simultaneous equations and . Find the value of .
Q12Challenging
A plumber charges $80 call-out plus $60 per hour. An electrician charges $50 call-out plus $75 per hour. After how many hours does the electrician become more expensive than the plumber? Give the number of hours at which they cost the same.
Worked solutions and answers at openmath.au/year-12/standard-2/types-of-relationships/simultaneous-linear-equations