Algebra and Graphs: Simultaneous linear equations

What you'll learn

Simultaneous linear equations appear frequently in CIE IGCSE Mathematics papers, typically worth 4-6 marks per question. This topic requires you to find values for two unknown variables that satisfy both equations at the same time, using either algebraic methods (elimination or substitution) or graphical interpretation. Strong competence in manipulating equations and systematic working is essential for success.

Key terms and definitions

Simultaneous equations — two or more equations that must be satisfied by the same values of the variables at the same time

Elimination method — an algebraic technique where one variable is removed by adding or subtracting equations after multiplying to create matching coefficients

Substitution method — an algebraic technique where one equation is rearranged to express one variable in terms of the other, then substituted into the second equation

Coefficient — the numerical factor multiplying a variable (e.g., in 3x + 2y = 7, the coefficient of x is 3)

Linear equation — an equation where variables appear only to the power of 1, producing a straight line when graphed

Solution pair — the values of x and y that satisfy both equations, written as (x, y)

Intersection point — the coordinates where two lines meet on a graph, representing the solution to simultaneous equations

Core concepts

Understanding simultaneous equations

Two linear equations with two unknowns typically have one unique solution. Consider:

• 2x + y = 8
• x - y = 1

The solution must work for both equations simultaneously. In CIE IGCSE Mathematics examinations, you encounter these in pure algebraic form, word problems requiring equation formation, or graphical questions.

Each equation represents a straight line when plotted. The solution corresponds to the point where the lines intersect. If lines are parallel (same gradient but different y-intercepts), no solution exists. If equations are multiples of each other (representing the same line), infinitely many solutions exist.

The elimination method

This method is typically faster and less prone to arithmetic errors, making it the preferred approach in examinations.

Step-by-step process:

1. Arrange equations in standard form (ax + by = c) with like terms aligned:

   3x + 2y = 13  ···· equation ①
   2x - y = 3    ···· equation ②
   

2. Identify the variable to eliminate. Look for coefficients that are easiest to match (often the variable with smallest coefficients).

3. Multiply one or both equations to create matching coefficients for the chosen variable:

   3x + 2y = 13  ···· equation ①
   4x - 2y = 6   ···· equation ② × 2
   

4. Add or subtract equations to eliminate that variable:

   7x = 19
   x = 19/7 (or as a mixed number or decimal if required)
   

5. Substitute back into either original equation to find the second variable:

   2(19/7) - y = 3
   38/7 - y = 3
   y = 38/7 - 21/7 = 17/7
   

6. Check your answer by substituting both values into the other equation.

When to add vs subtract:

• If the coefficients of the variable you're eliminating have opposite signs, add the equations
• If the coefficients have the same sign, subtract the equations

The substitution method

This method works well when one equation already has a variable isolated or can easily be rearranged.

Step-by-step process:

1. Rearrange one equation to make one variable the subject:

   y = 2x + 1  ···· equation ①
   3x + 2y = 20  ···· equation ②
   

2. Substitute this expression into the other equation:

   3x + 2(2x + 1) = 20
   

3. Expand and solve for the remaining variable:

   3x + 4x + 2 = 20
   7x = 18
   x = 18/7
   

4. Substitute back to find the other variable:

   y = 2(18/7) + 1 = 36/7 + 7/7 = 43/7
   

The substitution method proves particularly useful when:

• One equation is already in the form y = ... or x = ...
• One variable has a coefficient of 1
• The question explicitly asks you to use substitution

Graphical solution of simultaneous equations

CIE IGCSE Mathematics examinations regularly test your ability to solve simultaneous equations by drawing graphs.

Method:

1. Rearrange each equation into the form y = mx + c (gradient-intercept form)

2. Create a table of values for each equation, typically using at least three x-values

3. Plot both lines accurately on the same axes with appropriate scales

4. Identify the intersection point — read off the coordinates carefully

5. State the solution as x = ... and y = ..., or as the coordinate pair (x, y)

Examination requirements:

• Use a sharp pencil and ruler for straight lines
• Label each line clearly (equation ① and equation ②)
• Show your table of values or at least three plotted points per line
• Draw lines that extend across the full range of your axes
• Read intersection coordinates to the required accuracy (usually 1 decimal place or nearest 0.5)

Forming simultaneous equations from worded problems

Word problems require you to first translate information into algebraic equations before solving. This skill is heavily tested in CIE IGCSE examinations.

Systematic approach:

1. Define variables clearly: "Let x = ... and let y = ..."

2. Identify two separate pieces of information that create two different equations

3. Form equations using the relationships described

4. Solve using elimination or substitution

5. Answer the question in context with appropriate units

Common contexts in examinations:

• Cost problems (e.g., buying different quantities of items)
• Age problems
• Number problems (e.g., two numbers with given sum and difference)
• Geometry problems (e.g., angles, perimeter, area)
• Speed-distance-time scenarios

Special cases

Parallel lines (no solution):

If elimination produces a false statement like 0 = 5, the equations represent parallel lines with no intersection point. The system has no solution.

Example:

2x + 3y = 6
2x + 3y = 9

Subtracting gives 0 = 3, which is impossible.

Identical lines (infinite solutions):

If elimination produces a true statement like 0 = 0, the equations represent the same line. Every point on the line is a solution.

Example:

2x + 3y = 6
4x + 6y = 12  (equation ① × 2)

Worked examples

Example 1: Elimination method (typical 4-mark question)

Question: Solve the simultaneous equations:

5x + 2y = 16
3x - y = 5

Solution:

Label the equations:

5x + 2y = 16  ···· ①
3x - y = 5    ···· ②

Multiply equation ② by 2 to match the y-coefficients:

6x - 2y = 10  ···· ② × 2

Add to equation ① (signs are opposite, so we add):

5x + 2y = 16
6x - 2y = 10
___________
11x = 26

Therefore: x = 26/11

Substitute x = 26/11 into equation ②:

3(26/11) - y = 5
78/11 - y = 5
78/11 - 55/11 = y
y = 23/11

Answer: x = 26/11, y = 23/11 (or x = 2.36, y = 2.09 to 2 d.p.)

[Mark scheme: 1 mark for valid manipulation, 1 mark for x value, 1 mark for substitution method shown, 1 mark for y value]

Example 2: Word problem (typical 5-mark question)

Question: Adult tickets cost $x and child tickets cost $y. A family buys 2 adult tickets and 3 child tickets for $47. Another family buys 4 adult tickets and 1 child ticket for $53. Find the cost of each type of ticket.

Solution:

Let x = cost of adult ticket ($) Let y = cost of child ticket ($)

From the first family:

2x + 3y = 47  ···· ①

From the second family:

4x + y = 53   ···· ②

Multiply equation ① by 2:

4x + 6y = 94  ···· ① × 2

Subtract equation ②:

4x + 6y = 94
4x + y = 53
___________
5y = 41

Therefore: y = 41/5 = 8.2

Substitute into equation ②:

4x + 8.2 = 53
4x = 44.8
x = 11.2

Answer: Adult ticket costs $11.20, child ticket costs $8.20

[Mark scheme: 1 mark for defining variables, 1 mark for each correct equation, 1 mark for correct method, 1 mark for both answers with correct context]

Example 3: Substitution method (typical 4-mark question)

Question: Solve using substitution:

y = 3x - 2
2x + 5y = 34

Solution:

Substitute y = 3x - 2 into the second equation:

2x + 5(3x - 2) = 34

Expand brackets:

2x + 15x - 10 = 34

Collect terms:

17x = 44
x = 44/17

Substitute back into y = 3x - 2:

y = 3(44/17) - 2
y = 132/17 - 34/17
y = 98/17

Answer: x = 44/17, y = 98/17 (or x = 2.59, y = 5.76 to 2 d.p.)

[Mark scheme: 1 mark for correct substitution, 1 mark for x value, 1 mark for substitution to find y, 1 mark for y value]

Common mistakes and how to avoid them

• Mistake: Adding/subtracting equations without matching coefficients first. Students write 3x + 2y = 7 plus 2x - y = 4 and incorrectly get 5x + y = 11. Correction: Always multiply one or both equations first to create identical coefficients for the variable you're eliminating. Check your coefficients are exactly the same (including sign) before combining equations.

• Mistake: Forgetting to multiply every term when scaling equations. Students multiply 2x - y = 3 by 2 and write 4x - y = 6 (missing the y term). Correction: Use brackets when multiplying: 2 × (2x - y = 3) becomes 4x - 2y = 6. Multiply the entire equation including the constant term.

• Mistake: Sign errors when substituting negative values. If x = -3, substituting into 2x + y = 5 incorrectly gives -6 + y = 5, leading to y = 11 instead of -1. Correction: Always use brackets when substituting: 2(-3) + y = 5. Calculate step-by-step and check signs carefully.

• Mistake: Only finding one variable and forgetting to complete the solution. Students find x = 4 but don't substitute back to find y. Correction: A complete answer requires both values. Always substitute your first answer back into one original equation to find the second variable, then check by substituting both into the other equation.

• Mistake: Reading graph intersection points inaccurately, especially when points lie between grid lines. Correction: Use the grid squares to estimate carefully. If asked for 1 decimal place, use the scale to calculate: if the x-axis goes up in 2s and the point is halfway between 6 and 8, x = 7.0.

• Mistake: Misforming equations from word problems by confusing which quantities combine. Students might write x + y = 47 when the problem states "2 items cost $x each and 3 items cost $y each, total $47." Correction: Write out what you're defining clearly: "Let x = cost of one item". Then translate systematically: "2 lots of $x plus 3 lots of $y equals 47" becomes 2x + 3y = 47.

Exam technique for Algebra and Graphs: Simultaneous linear equations

• Show full working systematically. CIE IGCSE Mathematics mark schemes award method marks even if your final answer is wrong. Label equations as ① and ②, show your multiplication step clearly (e.g., "② × 3"), and write out the addition/subtraction explicitly. This structured approach makes method marks easily accessible.

• Choice of method matters for time efficiency. Unless specified, choose elimination when both equations are in standard form with similar-sized coefficients. Choose substitution when one variable is already isolated (y = ...) or has coefficient 1. Graphical methods take longer but may be required by the question wording.

• State your answer clearly in the required form. Look for command words: "Find the value of x and y" requires both values stated separately. "Solve" typically means the same. "Write down the coordinates" requires answer as (x, y). Always include units if the context involves money, length, time, etc.

• Check answers are reasonable. Substitute both values into one equation you haven't used for checking. In word problems, verify your answers make sense in context (e.g., costs should be positive, ages should be whole numbers if that's contextually appropriate). Many easy marks are lost through arithmetic slips that checking would catch.

Quick revision summary

Simultaneous linear equations require finding two variable values satisfying both equations. Elimination method: multiply equations to match coefficients, add or subtract to eliminate one variable, solve, then substitute back. Substitution method: rearrange one equation for a variable, substitute into the other, solve. Graphical method: plot both lines and read the intersection point coordinates. In word problems, define variables clearly and form two separate equations from given information. Always show systematic working, check answers, and state solutions in the required form for examination success.