For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.
How this chapter applies
Eclat core: substitution, elimination, graphical interpretation, and translating word problems into two linear equations form the main route.
School-sensitive extension: mixture, rate, and constraint models with several interpretation steps may exceed the common assessment depth.
2027 national comparison: K310 Topic N7 includes simultaneous linear equations in two variables by substitution, elimination, and graphical methods, plus formulating equations to solve problems.
Check your school: confirm whether graphical solutions, calculator checks, or exact algebraic working are required.
Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 05) Simultaneous Equations and Word Problems cover? A: Blend substitution and elimination to model two-variable contexts quickly and accurately.
The core idea is simple: Turn two unknowns into two equations, then remove one unknown at a time.
Use it as a working check: Use substitution when one variable is already isolated. Use elimination when matching coefficients can cancel a variable cleanly. Always translate the final numbers back into the story.
Then go one layer deeper: Follow the ticket and mixture examples to practise the full loop: define variables, write equations from the context, solve algebraically, and check whether the answer is realistic.
Simultaneous equations convert real-world statements into solvable algebra. Choose the method - substitution, elimination, or graphical - that minimises manipulation.
Rearrange one equation to make a variable the subject for substitution.
In elimination, line up coefficients so one variable cancels cleanly.
Always interpret the algebraic solution back in context to check feasibility.
Word-problem setup checkpoint
Before solving, separate the modelling work from the algebra. Most mistakes happen before the first elimination step.
Word-problem cue
Equation type to write
Check before solving
Total number, total mass, total volume, or total tickets
Count or amount equation, such as x+y=total
Both unknowns must use the same unit.
Total cost, revenue, value, or concentration amount
Weighted equation, such as price times quantity or percentage times volume
Coefficients must match the variable they multiply.
"One is ... more than the other"
Relationship equation, such as x=y+4
Translate the larger quantity carefully.
Fractions or percentages in a mixture
Amount of pure substance equation
Convert percentages to decimals or fractions before multiplying.
Misconception check: do not solve two equations until both describe the same two unknowns. If x changes meaning between the first and second equation, the system no longer models one situation.
Answer feasibility checkpoint
After solving, test the answer against the story before writing the final sentence. A pair of numbers can satisfy the algebra but still fail the context.
Context clue
Feasibility check
Final-answer move
Common trap
Counts of tickets, students, books, or bundles
Values should usually be whole numbers and non-negative.
State what each variable counts, such as 180 adult tickets and 180 student tickets.
Leaving the answer as (180,180) with no labels.
Money, revenue, or total value
Substitute both values back into the total-value equation.
Check the units and currency before rounding.
Rounding one variable early, then making the total miss the stated amount.
Mixture or concentration
Each volume or mass should be non-negative, and the final concentration should lie between the two starting concentrations.
Write both quantities with units and identify which concentration each one belongs to.
Reporting the right numbers but swapping the 20% and 50% solutions.
Lengths, ages, or time durations
Values should be positive and realistic for the wording.
Reject or review any negative value unless the story explicitly allows it.
Treating every algebraic solution as acceptable.
Worked check: if a ticket problem gives a=180 and s=180, the interpretation is not just "answer 180,180". It means 180 adult tickets and 180 student tickets. The check 48(180)+32(180)=14400 confirms the revenue.
Misconception check: solving the equations is not the final step in a word problem. The final answer must name the quantities, use the right units, and pass the context check.
Worked example - Ticket mix at a concert
Adult tickets cost $48 and student tickets cost $32. At a weekend concert, 360 tickets are sold for a total of $14400. How many of each ticket type were sold?
Let a be the number of adult tickets and s the number of student tickets. Translate the statements:
a+s=360.
48a+32s=14400.
Use elimination. Multiply the first equation by 32: 32a+32s=11520.
Subtract this from the revenue equation: (48a+32s)−(32a+32s)=14400−11520 giving 16a=2880
Solve for a: a=162880=180.
Substitute back into a+s=360: 180+s=360⇒s=180.
Quick check: 48×180+32×180=8640+5760=14400 which matches the stated revenue.
Therefore 180 adult tickets and 180 student tickets were sold.
Elimination alignment checkpoint
Before adding or subtracting equations, write down which variable is meant to disappear. Then make the coefficients match and choose the operation that actually cancels them.
Coefficients to cancel
Operation to use
Quick sign check
Same size, same sign, such as +2y and +2y
Subtract one whole equation from the other.
The y-terms become 2y−2y=0.
Same size, opposite signs, such as +3x and −3x
Add the two whole equations.
The x-terms become 3x+(−3x)=0
Different sizes, such as 2y and 5y
Multiply one or both equations first.
Both target coefficients should become the same size before cancellation.
Worked check: solve 3x+2y=22 and 5x+2y=34. The 2y terms have the same sign, so subtract the first equation from the second:
(5x+2y)−(3x+2y)=34−22
2x=12
Hence x=6. Substitute into 3x+2y=22: 18+2y=22, so y=2.
Misconception check: do not subtract only the first term. Elimination adds or subtracts the entire equation, so every term on both sides must follow the same operation.
Worked example - Substitution with an explicit subject
Solve the simultaneous equations y=2x−3 and 4x−5y=1.
Substitute y=2x−3 into the second equation: 4x−5(2x−3)=1.
Expand: 4x−10x+15=1.
Combine like terms: −6x+15=1.
Subtract 15 from both sides: −6x=−14.
Divide by −6: x=614=37
Substitute back for y: y=2(37)−3=314−39=35
Hence (x,y)=(37,35).
Worked example - Mixture model with elimination
A chemist needs 8 litres of a 35% acid solution. She mixes a 20% solution with a 50% solution. How many litres of each should she combine?
Let x be the volume (in litres) of the 20% solution and y the volume of the 50% solution.
Total volume equation: x+y=8.
Concentration equation: 0.20x+0.50y=0.35×8=2.8.
Multiply the concentration equation by 10 to clear decimals: 2x+5y=28.
Use elimination. Multiply the volume equation by 2: 2x+2y=16.
Subtract from the scaled concentration equation: (2x+5y)−(2x+2y)=28−16 gives 3y=12
Solve for y: y=4.
Substitute back: x+4=8⇒x=4.
She should mix 4 litres of the 20% solution with 4 litres of the 50% solution.
Practice Quiz
Translate word problems, choose a solving method, and check interpretations with quick feedback.
Try this
Solve
{2x+3y=19x−y=4
and explain what x and y represent if the equations model counts of two bundle types in a fund-raising sale.
