What are simultaneous equations in the context of IB MYP Mathematics?

Simultaneous equations are a set of two or more equations with multiple variables that are solved together, as they share common solutions. In the IB MYP Mathematics curriculum, students learn to solve these equations using various methods such as substitution, elimination, and graphical representation.

How do you solve simultaneous equations using the substitution method?

To solve simultaneous equations using the substitution method, you first solve one of the equations for one variable in terms of the other. Then, substitute this expression into the other equation. This will result in a single equation with one variable, which can be solved. Finally, substitute the value of this variable back into one of the original equations to find the value of the other variable.

What is the elimination method for solving simultaneous equations?

The elimination method involves adding or subtracting the equations to eliminate one of the variables. This is done by aligning the equations and manipulating them so that one variable cancels out. Once one variable is eliminated, you can solve for the remaining variable. After finding one variable, substitute it back into one of the original equations to find the other variable.

Can simultaneous equations be solved graphically, and if so, how?

Yes, simultaneous equations can be solved graphically by plotting each equation on a coordinate plane. The point where the graphs intersect represents the solution to the equations, as it is the set of values that satisfy both equations. This method provides a visual understanding of the solution, which is particularly useful in the IB MYP Mathematics curriculum.

What are some common applications of simultaneous equations in real life?

Simultaneous equations are used in various real-life applications such as calculating the point of intersection in supply and demand curves in economics, determining the best mix of ingredients in a recipe, or solving problems involving distances and speeds in physics. Understanding how to solve these equations is crucial for modeling and solving practical problems.

Why is "Making sign errors when subtracting equations." a common mistake in Simultaneous Equations?

Why it happens: Multi-step arithmetic. How to avoid it: Write each step on its own line. Use brackets around the equation you're subtracting to remember to flip all signs.

Why is "Finding one variable and forgetting to find the other." a common mistake in Simultaneous Equations?

Why it happens: Rush. How to avoid it: The solution is a PAIR. After finding x, always substitute to find y (or vice versa).

Why is "Not checking the answer in the second equation." a common mistake in Simultaneous Equations?

Why it happens: Trust the first. How to avoid it: Always check BOTH equations. A sign error in one step can give a wrong answer that fits the first equation by coincidence.

AlgebraIB MYP Maths Extended Level

Simultaneous Equations

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Simultaneous Equations — IB MYP Mathematics (Extended): solving two equations in two unknowns

When two unknowns satisfy two relationships, we can usually find them. This MYP Mathematics Extended note covers the three solution methods — elimination, substitution, and graphical — plus linear-quadratic systems.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Solve a system of linear equations by elimination or substitution.
MYP Mathematics A — Solve a linear-quadratic system.
MYP Mathematics C — Show clear algebraic working.
MYP Mathematics D — Model a real situation with a system of equations.

Elimination method

Match coefficients, then add/subtract to eliminate one variable.

Elimination works by combining the two equations so that one variable cancels.

Worked example. Solve the system: $\begin{cases} 2x + 3y = 13 \\ 4x - y = 5 \end{cases}$

Multiply second equation by 3 to match $y$ -coefficient: $12x - 3y = 15$ .
Add to first: $(2x + 3y) + (12x - 3y) = 13 + 15 \Rightarrow 14x = 28 \Rightarrow x = 2$ .
Substitute back: $2(2) + 3y = 13 \Rightarrow 3y = 9 \Rightarrow y = 3$ .
Solution: $(x, y) = (2, 3)$ .

Check by substituting in BOTH equations:

$2(2) + 3(3) = 4 + 9 = 13$ . ✓
$4(2) - 3 = 8 - 3 = 5$ . ✓

Strategy:

Decide which variable to eliminate (whichever is easier to match).
Multiply one or both equations to MAKE coefficients equal (in magnitude).
Add or subtract to eliminate.
Solve for the remaining variable.
Substitute back to find the other.
Check in BOTH original equations.

Match coefficients of one variable.
Add (opposite signs) or subtract (same signs) to eliminate.
Substitute back; check both equations.

Substitution method

Solve one for $x$ or $y$ , plug into the other.

Substitution works well when one equation is easily solved for one variable.

Worked example. Solve: $\begin{cases} y = 2x - 1 \\ 3x + 2y = 5 \end{cases}$

The first equation is already solved for $y$ . Substitute into the second:
$3x + 2(2x - 1) = 5$ .
$3x + 4x - 2 = 5$ .
$7x = 7 \Rightarrow x = 1$ .
Then $y = 2(1) - 1 = 1$ .
Solution: $(1, 1)$ .

When to choose substitution over elimination:

One equation is already in the form $y = \ldots$ or $x = \ldots$ .
The coefficients are large or awkward for elimination.
One equation is non-linear (then substitution is usually best).

Both methods work for any linear system — pick whichever looks easier.

Solve one equation for one variable.
Substitute into the other equation.
Solve for the remaining variable, then back-substitute.

Linear-quadratic systems (Extended)

Substitute the line into the parabola.

When one equation is linear and the other quadratic, substitution is the natural choice.

Worked example. Solve: $\begin{cases} y = x + 1 \\ y = x^2 - 1 \end{cases}$

Set the two expressions equal: $x + 1 = x^2 - 1$ .
Rearrange: $x^2 - x - 2 = 0$ .
Factorise: $(x - 2)(x + 1) = 0$ . So $x = 2$ or $x = -1$ .
Find corresponding $y$ values: $x = 2 \Rightarrow y = 3$ ; $x = -1 \Rightarrow y = 0$ .
Two intersection points: $(2, 3)$ and $(-1, 0)$ .

A linear-quadratic system can have 0, 1, or 2 solutions depending on whether the line crosses the parabola, is tangent to it, or misses it.

This corresponds to the discriminant of the resulting quadratic:

$\Delta > 0$ : 2 intersection points.
$\Delta = 0$ : line is tangent (1 point).
$\Delta < 0$ : no intersection.

Real context (criterion D). Two prices, two quantities, two constraints — system of equations is the natural model.

Substitute the linear $y = \ldots$ into the quadratic.
Solve the resulting quadratic.
0/1/2 intersection points possible.

How it’s examined

Criterion A: solve systems by elimination or substitution. Criterion C: clearly justify which method. Criterion D: model a real-context system.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Simultaneous Equations

Step-by-step solutions to past-paper-style questions on simultaneous equations, written exactly the way a tutor would explain them at the board.

1Elimination method
Building confidence• elimination
Question
Solve $\begin{cases} 3x + 2y = 16 \\ 5x - 2y = 16 \end{cases}$ .
Step-by-step solution
1. Step 1
  Add the equations: the $y$ terms cancel. $(3x + 5x) + (2y - 2y) = 32 \Rightarrow 8x = 32 \Rightarrow x = 4$ .
2. Step 2
  Substitute back into first: $3(4) + 2y = 16 \Rightarrow 2y = 4 \Rightarrow y = 2$ .
3. Step 3
  Check: $5(4) - 2(2) = 20 - 4 = 16$ . ✓
Answer
$x = 4$ , $y = 2$ .
2Substitution method
Building confidence• substitution
Question
Solve $\begin{cases} y = 3x - 4 \\ 2x + y = 11 \end{cases}$ .
Step-by-step solution
1. Step 1
  First equation gives $y = 3x - 4$ . Substitute into second.
2. Step 2
  $2x + (3x - 4) = 11 \Rightarrow 5x = 15 \Rightarrow x = 3$ .
3. Step 3
  Then $y = 3(3) - 4 = 5$ .
Answer
$x = 3$ , $y = 5$ .
3Linear-quadratic system
Stretch• linear-quadratic
Question
Find the intersection points of $y = x^2$ and $y = 4x - 3$ .
Step-by-step solution
1. Step 1
  Set expressions equal: $x^2 = 4x - 3$ .
2. Step 2
  Rearrange: $x^2 - 4x + 3 = 0$ .
3. Step 3
  Factorise: $(x - 1)(x - 3) = 0 \Rightarrow x = 1$ or $x = 3$ .
4. Step 4
  Find $y$ : $x = 1 \to y = 1$ ; $x = 3 \to y = 9$ .
Answer
$(1, 1)$ and $(3, 9)$ .
4Word problem
Stretch• application
Question
A cinema sells adult tickets at $12 and child tickets at $8. On Saturday they sold 50 tickets for $520. How many of each type?
Step-by-step solution
1. Step 1
  Let $a$ = adults, $c$ = children.
2. Step 2
  Tickets: $a + c = 50$ . Revenue: $12a + 8c = 520$ .
3. Step 3
  From first: $c = 50 - a$ . Substitute: $12a + 8(50 - a) = 520$ .
4. Step 4
  $12a + 400 - 8a = 520 \Rightarrow 4a = 120 \Rightarrow a = 30$ . Then $c = 20$ .
Answer
30 adults, 20 children.

Key Definitions and Keywords — Simultaneous Equations

Definitions to memorise and the exact keywords mark schemes credit for simultaneous equations answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Simultaneous equations
Examiner keyword
Two or more equations that must all hold true simultaneously.
Elimination method
Examiner keyword
Adding or subtracting equations (after matching coefficients) to remove one variable.
Substitution method
Examiner keyword
Solving one equation for one variable, then substituting into another.

Common Mistakes and Misconceptions — Simultaneous Equations

The traps other students keep falling into on simultaneous equations questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Making sign errors when subtracting equations.
Why it happens
Multi-step arithmetic.
How to avoid it
Write each step on its own line. Use brackets around the equation you're subtracting to remember to flip all signs.
✕Finding one variable and forgetting to find the other.
Why it happens
Rush.
How to avoid it
The solution is a PAIR. After finding $x$ , always substitute to find $y$ (or vice versa).
✕Not checking the answer in the second equation.
Why it happens
Trust the first.
How to avoid it
Always check BOTH equations. A sign error in one step can give a wrong answer that fits the first equation by coincidence.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

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Simultaneous Equations — frequently asked questions

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Simultaneous Equations

Short Study Notes in the form of Slides

Short Study Notes — Simultaneous Equations

Detailed Notes

What you’ll learn

How it’s examined

1Elimination method

2Substitution method

3Linear-quadratic system

4Word problem

Simultaneous equations

Elimination method

Substitution method

✕Making sign errors when subtracting equations.

✕Finding one variable and forgetting to find the other.

✕Not checking the answer in the second equation.

Practice questions

Past paper style quiz

Assess your understanding

Simultaneous Equations — frequently asked questions