Systems of Linear Equations

Introduction: Why do systems matter? 🌍

Imagine you are planning a school event, students. You need to know how many tickets to sell, how much food to buy, or how to split costs between students and parents. Often, one equation is not enough. You may have two or more unknown quantities, and several conditions that all must be true at the same time. That is exactly what a system of linear equations helps you do.

A system of linear equations is a set of two or more linear equations that use the same variables. The goal is to find values that make every equation true at once. In IB Mathematics: Applications and Interpretation HL, this topic connects to real-life modelling, technology, graphs, and algebraic reasoning. Systems also fit naturally into the broader area of Number and Algebra because they use patterns, algebraic structure, and exact numerical relationships.

Learning objectives

By the end of this lesson, you should be able to:

explain the main ideas and terminology behind systems of linear equations,
solve systems using algebraic and graphical methods,
use technology to interpret solutions,
connect systems to applications in finance, planning, and data modelling,
summarize how systems fit into the Number and Algebra topic.

1. What is a system of linear equations?

A linear equation is an equation whose graph is a straight line. In two variables, it often looks like $ax+by=c$, where $a$, $b$, and $c$ are constants and $x$ and $y$ are variables. A system means multiple equations considered together.

For example:

$\begin{cases}$

$2x+y=7\\$

$x-y=1$

$\end{cases}$

This system asks for a pair of values, $x$ and $y$, that satisfy both equations at the same time.

Key terminology

Variable: a symbol such as $x$ or $y$ that stands for an unknown value.
Coefficient: the number multiplying a variable, such as $2$ in $2x$.
Solution: values of the variables that make all equations in the system true.
Consistent system: a system with at least one solution.
Inconsistent system: a system with no solution.
Independent system: a system with exactly one solution.
Dependent system: a system with infinitely many solutions, usually because the equations represent the same line.

A useful way to think about a system is as a set of rules. The solution must obey every rule at once. In real life, this happens when two conditions must be satisfied together, such as a budget limit and a purchase total 💡.

Example

Suppose a cinema sells adult tickets for $a$ dollars and student tickets for $s$ dollars. If 2 adult tickets and 1 student ticket cost $29$, and 1 adult ticket and 2 student tickets cost $28$, then

$\begin{cases}$

$2a+s=29\\$

$a+2s=28$

$\end{cases}$

Solving this system tells us the ticket prices.

2. Solving systems algebraically

IB AI HL expects you to use algebra efficiently and to choose a method that fits the situation. Two common methods are substitution and elimination.

Substitution method

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

Example:

$\begin{cases}$

$y=3x-5\\$

$2x+y=7$

$\end{cases}$

Since $y=3x-5$, substitute that into the second equation:

$2x+(3x-5)=7$

Simplify:

$5x-5=7$

Add $5$ to both sides:

$5x=12$

So,

$ x=\frac{12}{5}$

Now substitute back into $y=3x-5$:

$ y=3\left(\frac{12}{5}\right)-5=\frac{36}{5}-\frac{25}{5}=\frac{11}{5}$

So the solution is

$\left(\frac{12}{5},\frac{11}{5}\right)$

Elimination method

This method is useful when coefficients line up nicely, because you can add or subtract equations to remove one variable.

Example:

$\begin{cases}$

$3x+2y=16\\$

$3x-2y=8$

$\end{cases}$

Add the equations:

$6x=24$

So,

$ x=4$

Substitute into $3x+2y=16$:

$3(4)+2y=16$

Then

$12+2y=16$

So,

$2y=4$

and

$ y=2$

The solution is $(4,2)$.

Why algebra matters

Algebra gives exact answers when the system has neat values or fractions. It also helps you see structure. For example, if two equations become identical after simplifying, then the system has infinitely many solutions. If the variables cancel and produce something false like $0=5$, the system has no solution.

3. Graphical interpretation and meaning

Graphing is not just about drawing lines. It gives a visual meaning to the solution. The point where two lines intersect is the ordered pair that satisfies both equations.

For the system

$\begin{cases}$

$y=2x+1\\$

$y=-x+7$

$\end{cases}$

the lines cross at one point. To find it algebraically, set the equations equal because both equal $y$:

$2x+1=-x+7$

Add $x$ to both sides and subtract $1$:

$3x=6$

$ x=2$

Then

$ y=2(2)+1=5$

The intersection is $(2,5)$.

What graphs tell us

If two lines intersect once, there is one solution.
If two lines are parallel, there is no solution.
If two lines are the same line, there are infinitely many solutions.

This is especially useful when data is approximate. In real-world modelling, you may not get exact values, so graphing helps you interpret whether a model is reasonable 📈.

Technology-supported interpretation

In IB AI HL, technology is important. Graphing calculators, spreadsheet software, and dynamic graphing tools can help you:

plot several lines quickly,
estimate intersections,
check algebraic answers,
study how changing coefficients changes the solution.

For example, if one line becomes steeper by changing $y=mx+b$, you can observe how the intersection point shifts. This is part of mathematical modelling: changing one assumption changes the outcome.

4. Systems in real-life modelling

Systems of linear equations appear in many practical situations. They are especially useful when there are two unknowns and two conditions.

Finance example

A phone plan has a fixed monthly fee plus a cost per gigabyte. Let $f$ be the fixed fee and $g$ the cost per gigabyte. If 3 GB costs $41$ and 7 GB costs $53$, then

$\begin{cases}$

$f+3g=41\\$

$f+7g=53$

$\end{cases}$

Subtract the first equation from the second:

$4g=12$

$ g=3$

Then

$ f+3(3)=41$

which gives

$ f=32$

So the model is a fixed fee of $32$ and a rate of $3$ dollars per GB.

Mixture example

Suppose a lab needs $10$ liters of a solution that is $20\%$ acid and $50\%$ acid. If $x$ liters of the $20\%$ solution and $y$ liters of the $50\%$ solution are mixed, then

$\begin{cases}$

$x+y=10\\$

$0.2x+0.5y=3.5$

$\end{cases}$

The first equation counts total volume, and the second counts pure acid. Systems like this help scientists, chemists, and engineers make accurate mixtures.

Why this is Number and Algebra

These examples show that systems are not isolated skills. They connect to:

algebraic representation, because the situation becomes equations,
numerical modelling, because the equations represent real quantities,
problem solving, because you must interpret the result in context,
technology-supported interpretation, because graphs and calculators help verify the model.

5. Systems with more than two equations or variables

A system can also involve three variables, such as $x$, $y$, and $z$. For example:

$\begin{cases}$

$x+y+z=12\\$

$2x-y+z=7\\$

$x+2y-z=5$

$\end{cases}$

These systems may be solved by elimination, substitution, or matrices when appropriate. In AI HL, you should understand the logic of solving them and be able to interpret the number of solutions.

Possible outcomes

One unique solution: one exact set of values satisfies every equation.
No solution: the equations contradict each other.
Infinitely many solutions: the equations describe the same constraint in different forms.

In a geometric sense, three-variable linear equations represent planes in three-dimensional space. The common solution can be a point, a line, a plane, or nothing, depending on how the planes intersect.

Even when technology solves a system quickly, you still need to understand what the answer means. A calculator can compute a result, but only you can explain whether it makes sense in the situation.

Conclusion

Systems of linear equations are a core part of Number and Algebra because they combine algebraic manipulation, numerical reasoning, and interpretation. They help solve realistic problems where several conditions must be true at the same time. Whether you use substitution, elimination, graphing, or technology, the central idea is always the same: find the values that satisfy every equation in the system.

For IB Mathematics: Applications and Interpretation HL, this topic is important because it links pure algebra to practical modelling. students, when you study systems, focus on the structure of the equations, the meaning of the solution, and the reasonableness of the result in context. That combination of algebra and interpretation is a major part of success in this course ✨.

Study Notes

A system of linear equations is a set of linear equations using the same variables.
A solution is a value or set of values that satisfies every equation in the system.
A system can be consistent with one solution, no solution, or infinitely many solutions.
Substitution is useful when one variable is already isolated or easy to isolate.
Elimination is useful when adding or subtracting equations removes one variable.
Graphically, the solution is the intersection of the lines.
Parallel lines mean no solution; the same line means infinitely many solutions.
Technology helps check answers, estimate intersections, and explore changing parameters.
Systems appear in finance, mixtures, planning, and other real-world models.
In Number and Algebra, systems connect algebraic structure, modelling, and interpretation.