Simultaneous Equations & Linear Inequalities
Equation Pairs to Solutions — Simple
Draw a line from each pair of simultaneous equations to its solution (x, y).
Substitution or Elimination?
For each pair of equations, circle the most efficient method to solve them.
y = 3x + 1 and 2x + y = 11
2x + 3y = 12 and 2x − y = 4
y = x − 5 and y = 2x + 1
3x + 2y = 14 and 5x − 2y = 10
Identify the Variables
For each word problem, circle the correct choice of variables.
Adult tickets cost $15 and child tickets cost $10. A group buys some of each.
A shop sells pens and notebooks. You know the total items and total cost.
Two friends have a combined age of 30 and one is 4 years older.
Method Name to Description
Draw a line from each solving method to its description.
Consistent vs Inconsistent Systems
Sort each system of equations: Consistent (has a solution) or Inconsistent (no solution).
Does the Point Satisfy Both Equations?
Circle YES if the given point satisfies both equations, or NO if it does not.
Point (3, 5): y = 2x − 1 and x + y = 8
Point (1, 4): y = 3x + 2 and 2x + y = 7
Point (2, 1): x − y = 1 and 3x + 2y = 8
Point (4, 3): y = x − 1 and y = 2x − 6
Number of Solutions
Circle the correct number of solutions for each system of equations.
y = 2x + 3 and y = −x + 6 (different gradients)
y = 3x + 1 and y = 3x − 4 (same gradient, different intercept)
2x + 4y = 8 and x + 2y = 4 (one is a multiple of the other)
y = −x + 5 and y = x − 1 (different gradients)
Graph Description to Number of Solutions
Sort each graph description by the number of solutions the system has.
Correct Substitution Step
For each equation pair, circle the correct substitution step.
y = 3x + 2 and y = x + 8 → Set equal:
y = 4x − 3 and 2x + y = 9 → Substitute y:
x = 2y + 1 and 3x − y = 8 → Substitute x:
y = x − 5 and y = −2x + 7 → Set equal:
Correct Elimination Step
For each pair of equations, circle the correct elimination step.
2x + y = 10 and x − y = 2 → Add equations:
x + 2y = 7 and 3x + 2y = 13 → Subtract first from second:
3x + 4y = 18 and 3x − 2y = 6 → Subtract second from first:
5x + 3y = 21 and 2x + 3y = 12 → Subtract second from first:
Solve by Substitution — Step by Step
Put the steps in the correct order to solve: y = 2x − 1 and y = −x + 5
Solve by Elimination — Step by Step
Put the steps in the correct order to solve: 3x + 2y = 16 and x − 2y = 0
Solve and Find x
Circle the correct value of x for each system.
x + y = 10 and x − y = 4 → Add equations: 2x = 14, so x =
y = x + 3 and y = 2x − 1 → Set equal: x + 3 = 2x − 1, so x =
2x + 3y = 19 and 2x + y = 11 → Subtract: 2y = 8, y = 4, so 2x + 4 = 11, x =
3x − y = 7 and x + y = 5 → Add: 4x = 12, so x =
Find y After Finding x
Given the value of x, circle the correct value of y.
x = 7 in x + y = 10:
x = 4 in y = x + 3:
x = 3.5 in 2x + y = 11:
x = 3 in x + y = 5:
Word Problem to Equation Pair
Draw a line from each word problem to the correct pair of simultaneous equations.
Linear Inequality Notation
Circle the correct inequality that represents each statement.
x is greater than or equal to 3
y is less than 2x + 1
The sum of x and y is at most 10
x is strictly between −2 and 5
Which Side of the Line to Shade?
For each inequality, circle the correct region to shade on a graph.
y < x + 3
y ≥ −2x + 4
x + y > 6
y ≤ 3 (horizontal line)
Points Inside vs Outside the Region
The feasible region is defined by y ≤ x + 4 and y ≥ 0 and x ≥ 0. Sort each point: Inside or Outside the region.
Solve Algebraically — System 1
Solve the following system of simultaneous equations. Show all working and verify your answer.
Solve: 3x + 2y = 16 and 5x − 2y = 8. Use an appropriate method (substitution or elimination). Show every step and check your answer by substituting back into both equations.
Solve Algebraically — System 2
Solve the following system of simultaneous equations. Show all working and verify your answer.
Solve: 2x + 5y = 21 and 3x − 2y = 4. (Hint: you may need to multiply one or both equations before eliminating.) Show every step and check your answer.
Word Problem — Ticket Prices
Write simultaneous equations for the problem, then solve. Show all working.
Movie tickets cost $12 for adults and $8 for children. A family bought 6 tickets for $56. Let a = number of adult tickets and c = number of child tickets. Write two equations, solve the system, and state how many of each ticket type were purchased.
Word Problem — Phone Plans
Write simultaneous equations for the problem, then solve. Show all working.
A phone plan charges a monthly fee (f) plus a per-gigabyte rate (g). Using 5 GB costs $35 total, and using 12 GB costs $56 total. Write two equations, solve for f and g, and find the cost of using 20 GB in a month.
Graph Two Inequalities
Graph the following system of linear inequalities and describe the feasible region.
On a set of axes, graph the region satisfying: y ≤ 2x + 4, y ≥ −x + 1, and x ≥ 0. Label the boundary lines, shade the feasible region, and find the coordinates of the vertices (corner points) of the region.
Describe the shape of the feasible region and explain how you determined which side of each line to shade.
Write Equations from a Word Problem
Translate the following situation into a system of equations. You do NOT need to solve — just define variables and write the equations.
A bakery sells cupcakes for $4 each and slices of cake for $6 each. On Saturday they sold 45 items for a total of $210. Define appropriate variables and write two simultaneous equations that model this situation.
A farmer has chickens and cows. In total there are 30 animals and 86 legs. Define appropriate variables and write two simultaneous equations that model this situation.
True or False — Simultaneous Equations & Inequalities
Circle TRUE or FALSE for each statement.
Two linear equations with different gradients always have exactly one solution
If two lines have the same gradient and same y-intercept, they have no solutions
The solution to a system of two linear equations is always a point (x, y)
When graphing y < 3x + 1, the boundary line should be dashed (not solid)
Multiplying both sides of an inequality by a negative number reverses the inequality sign
Create Your Own Word Problem
Design your own simultaneous equations word problem.
Write a real-world word problem that can be modelled by a system of two simultaneous linear equations. Then provide the equations and solve your own problem, showing all working. Make sure the solution gives whole-number answers.
Equations in Everyday Life
Find real-world situations that involve simultaneous equations.
- 1Compare two mobile phone plans with different monthly fees and per-minute rates. Set up simultaneous equations and find the number of minutes where both plans cost the same.
- 2At a cafe, price two different breakfast combos (e.g. 2 coffees + 1 muffin = $14, 1 coffee + 2 muffins = $13). Write equations and figure out the individual prices.
- 3Research two gym memberships with different joining fees and monthly rates. Set up equations and calculate the number of months at which they break even.
Graphical Exploration
Explore simultaneous equations and inequalities graphically.
- 1Use a free graphing tool (like Desmos) to graph two linear equations simultaneously. Observe where the lines intersect and verify the solution algebraically.
- 2Graph the inequalities y ≤ x + 3 and y ≥ −x + 1 in Desmos. Describe the shape and size of the shaded region. What happens if you change the constants?
- 3Create a system of two parallel lines in Desmos (same gradient, different y-intercept). Explain why the system has no solution by observing the graph.
Substitution Method — Step by Step
Solve each pair of simultaneous equations using the substitution method.
Solve: y = 2x − 1 and 3x + y = 14. Show all steps: substitute, simplify, solve for x, then find y. Verify by substituting back.
Solve: y = 4 − x and 2x − y = 2. Show all steps and verify your solution.
Solve: x = 3y + 5 and 2x − y = 15. Show all steps and verify.
Check a Solution — Simultaneous Equations
Circle whether each given solution is correct or incorrect.
x + y = 7 and x − y = 3. Solution: x = 5, y = 2
2x + y = 10 and x + 3y = 15. Solution: x = 3, y = 4
y = 2x + 1 and y = x + 5. Solution: x = 4, y = 9
3x − 2y = 4 and x + y = 7. Solution: x = 2, y = 5
Word Problem to Equation Pair
Draw a line from each word problem to the pair of equations that models it.
Elimination Method
Solve each pair using the elimination method.
Solve: 3x + 2y = 16 and 3x − y = 7. Subtract equations to eliminate x, then solve. Verify.
Solve: 2x + 5y = 23 and 4x + 5y = 31. Subtract to eliminate 5y, find x, then y. Verify.
Solve: 3x + 4y = 25 and 2x + 3y = 18. Multiply to make coefficients match before eliminating.
One, Zero, or Infinite Solutions?
Sort each pair of simultaneous equations.
Linear Inequalities — Solve and Interpret
Solve and interpret linear inequalities.
Solve 3x − 5 > 7 and represent the solution on a number line. Explain what the open/closed circle means.
Solve −2x + 3 ≤ 11. Show all algebraic steps. What is the key rule for multiplying or dividing by a negative number?
Solve 2 ≤ 3x + 5 < 14 (compound inequality). Write the solution as an interval.
Inequality Direction — Which Way?
Circle the correct inequality symbol after the transformation.
Starting with x < 5: multiply both sides by −2:
Starting with −3x ≥ 12: divide both sides by −3:
Starting with x + 7 > 3: subtract 7:
Starting with 4x < −8: divide both sides by 4:
Simultaneous Equations — Real World
Set up and solve simultaneous equations for real-world problems.
A cinema charges $15 for adults and $9 for children. 200 people attend and total takings are $2,220. How many adults and children attended? Define variables, write equations, solve, and verify.
A small business sells handmade mugs for $25 and plates for $18. In one day they sell 30 items and take in $654. How many of each did they sell?
Elimination Method Steps — Order Them
Put the steps for elimination method in the correct order.
Inequality Solution to Graph Description
Draw a line from each inequality solution to its correct number-line description.
Break-Even Analysis
Use simultaneous equations to find a business break-even point.
A candle-making business has fixed costs of $300 per month and variable costs of $4 per candle. They sell candles for $12 each. (a) Write equations for total cost C and total revenue R in terms of quantity q. (b) Set C = R and solve to find the break-even quantity. (c) How much profit do they make if they sell 60 candles in a month?
Algebraic vs Graphical Method
Sort each situation: better solved Algebraically or Graphically.
Compound Inequalities in Context
Solve and interpret compound inequalities in real-world contexts.
A company needs between 50 and 120 items manufactured today (inclusive). If the machine produces x items per hour and runs for 8 hours, write and solve a compound inequality to find the required rate x.
Temperature for a chemistry experiment must be between 15°C and 25°C. The thermostat reads T = 2h + 11 where h is hours since start. For which values of h is the temperature in the safe range?
Methods Used to Solve Simultaneous Equations
Tally which method was used across 30 practice problems.
| Item | Tally | Total |
|---|---|---|
Substitution method | ||
Elimination method | ||
Graphical method | ||
Technology/CAS |
Interpret Solutions — Simultaneous Equations
Circle the correct interpretation of each solution.
Cost: C = 50 + 20n, Revenue: R = 35n. Solving gives n = 10. Explain what this means in context — what has happened at n = 10, and what does profit look like before and after that point?
Two trains leave stations 300 km apart, travelling toward each other. One travels at 80 km/h, the other at 100 km/h. Write equations for position over time and solve to find when and where they meet.
Linear Inequalities — Two Variables
Describe the solution region for a linear inequality in two variables.
For y > 2x − 3, describe: (a) the boundary line, (b) whether the line is solid or dashed, (c) which side is shaded. Test the point (0, 0) to confirm.
For y ≤ −x + 5 and y ≥ x − 1, describe the region that satisfies both inequalities. Find two points in this region and verify they satisfy both inequalities.
Solutions of Simultaneous Equations — Graphical Meaning
Circle the correct graphical meaning for each type of solution.
One solution (x, y): the two lines
No solution: the two lines
Infinite solutions: the two lines
Mixture Problems — Simultaneous Equations
Use simultaneous equations to solve mixture problems.
A chemist needs to make 10 litres of a 30% acid solution by mixing a 20% acid solution with a 50% acid solution. (a) Let x = litres of 20% solution, y = litres of 50% solution. Write two equations. (b) Solve the system and find the exact quantities needed. (c) Verify your answer by checking that the total acid content is correct.
Classify the System — Solution Type
Sort each system of equations: Consistent (one solution), Inconsistent (no solution), or Dependent (infinite solutions).
Linear Programming Introduction
Use systems of inequalities to model a simple linear programming problem.
A bakery makes muffins (profit $3 each) and cakes (profit $7 each). Constraints: • Total items ≤ 50 per day • At least 10 muffins must be made • At least 5 cakes must be made Let m = muffins, c = cakes. Write the constraint inequalities. Find the corner points of the feasible region. Determine how many of each to maximise profit.
Simultaneous Equations — Applications to Areas
Draw a line from each type of simultaneous equation problem to its application area.
Distance-Rate-Time — Simultaneous Equations
Model distance-rate-time problems using simultaneous equations.
Two cyclists start from opposite ends of an 80 km trail at the same time. Cyclist A rides at 20 km/h and Cyclist B at 12 km/h. Set up two equations for their positions and find when and where they meet.
A boat travels 60 km upstream in 5 hours and returns in 3 hours. Find the speed of the boat in still water and the speed of the current. (Hint: upstream speed = boat speed − current, downstream = boat speed + current)
Inequality Word Problems — Set Up Correctly
Circle the inequality that correctly models each situation.
A student needs at least 60% to pass. Their test score is s:
A lift can carry at most 12 people. The number n of people in the lift:
A temperature must remain below 25°C. Temperature T:
To qualify, a runner must finish in less than 45 minutes. Time t:
Simultaneous Equations — Real Data
Find and solve a real simultaneous equations problem from your life.
- 1Compare the costs of two mobile phone plans. Plan A charges $30/month + $0.10/text, Plan B charges $45/month with unlimited texts. Set up and solve the simultaneous equations to find at which number of texts both plans cost the same.
- 2Visit two different stores (or their websites) and find two items sold at both. Note the prices. If you could buy x units of item 1 and y units of item 2 with a fixed budget from each store, set up simultaneous equations to find how many you could buy.
- 3Research electricity plans in your state. If Plan A charges 30c/kWh with a $1/day supply charge, and Plan B charges 25c/kWh with a $1.50/day supply charge, find the daily usage at which the costs are equal.
Verify Simultaneous Equation Solutions
Verify solutions by substituting back into both original equations.
A student claims the solution to: 5x + 2y = 19 and 3x − y = 8 is x = 3, y = 2. (a) Substitute into both equations and check if the solution is correct. (b) If it is wrong, find the correct solution using either substitution or elimination. (c) Explain what 'checking a solution' means mathematically.
Equation Types Encountered in Algebra
Tally the types of equations you encounter in a week of maths study.
| Item | Tally | Total |
|---|---|---|
Linear equations (one variable) | ||
Linear equations (two variables — simultaneous) | ||
Linear inequalities | ||
Quadratic equations | ||
Exponential equations |
Break-Even Analysis
Use simultaneous equations to solve business break-even problems.
A business has fixed costs of $2,000/month and variable costs of $8 per unit. Revenue is $15 per unit. Write equations for total cost C and total revenue R in terms of units sold x.
Find the break-even point by solving C = R. How many units must be sold to break even?
Graph both equations and shade the profit region. Label the break-even point.
If fixed costs rise to $2,500, how does the break-even point change? Calculate the new break-even.
Solving Methods — Match to Best Use
Match each solving method to the type of simultaneous equations it is best suited to.
Linear Programming
Formulate and solve a simple linear programming problem.
A factory makes tables (T) and chairs (C). Each table takes 3 hours and each chair takes 1 hour. Available hours: 12. Write the time constraint as an inequality.
At least 2 tables and 2 chairs must be made. Write these as inequalities.
Profit: $50 per table, $20 per chair. Write the objective function P = ....
Graph the feasible region and test each corner point. State the production plan that maximises profit.
Inequalities — Match the Graph Description
Match each inequality to its graph description on a number line.
Non-Linear Simultaneous Equations
Solve systems where one equation is linear and one is quadratic.
Solve simultaneously: y = x + 2 and y = x². Find all intersection points algebraically.
Interpret your answer geometrically: how many times does the parabola y = x² cross the line y = x + 2?
Solve simultaneously: y = 2x − 1 and y = x² − 3. Show all working.
Can a line and parabola intersect at exactly one point? Explain and give an example.