Systems of Linear Equations — I can solve systems of linear...

SECTION 1

Where Did Systems of Equations Come From?

People have been solving problems with two unknowns for thousands of years. Imagine a farmer in ancient China who knows the total number of chickens and rabbits in a pen, plus the total number of legs. How many of each animal are there? That is exactly the kind of puzzle a system of equations can solve.

Over the centuries, mathematicians from many cultures found clever ways to handle two or more equations at the same time. Let's look at a few key moments in that story.

200 BCE

Ancient China — The Nine Chapters

The Chinese textbook Nine Chapters on the Mathematical Art showed how to solve problems with multiple unknowns using a method similar to elimination.

820 CE

Al-Khwarizmi's Algebra

The Persian mathematician Al-Khwarizmi wrote one of the first algebra books. His work gave us the word "algebra" and laid the groundwork for solving equations with variables.

1637

Descartes Invents Coordinate Graphing

René Descartes connected algebra and geometry by creating the coordinate plane (the x-y grid). This made it possible to see equations as lines on a graph.

Today

Systems Everywhere

Systems of equations power GPS navigation, video-game physics, business planning, and much more. Whenever two conditions must be true at the same time, a system of equations can help.

The big question these mathematicians were asking is still the question we ask today: When two rules apply at the same time, what values make both rules true?

SECTION 2

Core Ideas You Need to Know

Before we start solving, let's nail down a few important ideas. These are the building blocks for everything else in this lesson.

Linear Equation

An equation whose graph is a straight line. It looks like y = mx + b, where m is the slope and b is the y-intercept.

System of Equations

Two (or more) equations that share the same variables. You need to find values of x and y that work in both equations at the same time.

Solution of a System

The ordered pair (x, y) that makes every equation in the system true. On a graph, it is the point where the lines cross (intersect).

Substitution

A solving method where you replace one variable in an equation with an expression from the other equation. Think of it as swapping in a clue.

Intersection Point

The exact spot on the coordinate plane where two lines meet. Its coordinates (x, y) are the solution to the system.

✦ KEY TAKEAWAY

Think of two friends walking along different paths in a park. Each path is a line. If the paths cross, the friends will meet at exactly one spot. That meeting spot is the solution of the system — the one (x, y) point that lies on both paths.

SECTION 3

Seeing the Solution on a Graph

One of the best ways to understand a system of equations is to graph both lines and look for where they meet. The diagram below shows two lines on the same coordinate plane. Notice the point where they cross — that is the solution!

The cyan line represents y = x, and the violet line represents y = −x + 6. They intersect at the pink dot (3, 3), which is the solution of the system.

In the diagram above, the cyan line climbs upward and the violet line slopes downward. They meet at (3, 3). If you plug x = 3 into both equations, you get y = 3 each time. That confirms (3, 3) is the solution.

✔️ Quick Check

Equation 1: y = x → y = 3 ✓ Equation 2: y = −x + 6 → y = −3 + 6 = 3 ✓ Both true, so (3, 3) is correct!

SECTION 4

The Math Behind Solving a System

There are different ways to solve a system of linear equations. In this lesson we will focus on two beginner-friendly methods: graphing and substitution. Both lead to the same answer.

Method 1 — Graphing

Graph each equation on the same coordinate plane. The point where the lines cross is the solution. This method is great for building understanding, but it can be hard to read exact answers if the intersection falls between grid lines.

Method 2 — Substitution

With substitution, you use one equation to express a variable, then plug that expression into the other equation. Here is the general idea.

STARTING SYSTEM

y = mx + b and y = nx + c

m and n are slopes; b and c are y-intercepts. Both equations equal y, so we can set the right sides equal to each other.

SET RIGHT SIDES EQUAL

mx + b = nx + c

Since both expressions equal y, they must equal each other at the intersection point. Solve this equation for x.

SOLVE FOR x

x = (c − b) ÷ (m − n)

Once you know x, substitute it back into either original equation to find y. The pair (x, y) is your solution.

💡 Don't Worry!

That formula looks fancy, but for simple numbers you usually just do it step by step. You will see a full worked example in Section 6.

SECTION 5

Three Types of Systems

Not every system has exactly one solution. Depending on how the lines relate to each other, there are three possible outcomes. The diagram below shows all three.

Left: The lines cross at one point — one solution. Center: The lines are parallel — no solution. Right: The lines are the same — infinitely many solutions.

Summary of system types

Type	What the Lines Do	Number of Solutions
Intersecting	Cross at one point	Exactly one
Parallel	Same slope, different y-intercepts	None (no solution)
Same Line	Same slope AND same y-intercept	Infinitely many

In this lesson we focus on the first type — two lines that cross at exactly one point. That crossing point is the answer we are looking for.

SECTION 6

Worked Example: Solving Step by Step

Let's solve a system using substitution. Here is the problem:

SYSTEM TO SOLVE

y = 2x + 1 and y = −x + 7

Find the values of x and y that satisfy both equations.

Solving y = 2x + 1 and y = −x + 7

Step 1 — Notice Both Equations Equal yBoth equations are already solved for y. Since the left sides are both y, the right sides must be equal at the intersection point.

Step 2 — Set the Right Sides EqualWrite: 2x + 1 = −x + 7. This is one equation with one unknown (x), which we can solve.

2x + 1 = −x + 7

Step 3 — Solve for xAdd x to both sides: 2x + x + 1 = 7, so 3x + 1 = 7. Subtract 1 from both sides: 3x = 6. Divide both sides by 3: x = 2.

x = 2

Step 4 — Substitute x Back to Find yPick either equation. Using y = 2x + 1: y = 2(2) + 1 = 4 + 1 = 5.

y = 5

Step 5 — Check in the Other EquationPlug x = 2 into y = −x + 7: y = −2 + 7 = 5. It matches! Both equations give y = 5 when x = 2.

Solution: (2, 5) ✓

💡 Pro Tip

Always check your answer by plugging (x, y) into both original equations. If one doesn't work, go back and look for mistakes.

SECTION 7

Graphing vs. Substitution — Pros and Cons

Both methods get you to the same answer, but each has strengths and weaknesses. Here is a side-by-side look.

Comparison of graphing and substitution methods

Feature	Graphing	Substitution
Visual?	Yes — you can see the lines and their intersection	No — it is all algebra
Exact answer?	Only if the intersection lands on a grid point	Always exact
Speed	Slower (draw two lines)	Usually faster for simple equations
Best for	Understanding what a solution looks like	Getting a precise numeric answer

✦ KEY TAKEAWAY

Graphing is like using a map to find where two roads meet — you see the big picture. Substitution is like using GPS coordinates — you get the exact spot. Both are useful tools in your math toolkit!

SECTION 8

What Comes Next?

You are building skills now that will power bigger ideas later. In Algebra 1 and beyond, you will learn more methods and tackle tougher systems. Here is a preview.

Your current skills compared with future topics

What You Know Now	What You'll Learn Later
Substitution (swap one variable)	Elimination (add or subtract whole equations)
Systems of two equations	Systems of three or more equations
Lines (linear equations)	Curves — parabolas, circles, and more
Graphing by hand	Matrices and technology-aided solving

The core idea stays the same no matter how advanced the math gets: find the values that satisfy every equation at the same time. Mastering simple systems now gives you a strong foundation for everything ahead.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get trickier. After each one, check your work by reading the answer.

PROBLEM 1 — CONCEPTUAL

Two lines are graphed on the same coordinate plane. Line A passes through (0, 1) and (2, 5). Line B passes through (0, 7) and (2, 5). What is the solution of the system? How do you know?

PROBLEM 2 — BASIC CALCULATION

Solve the system: y = x + 3 and y = 2x + 1. Use substitution.

PROBLEM 3 — INTERMEDIATE

Solve: y = 3x − 4 and y = −x + 8.

PROBLEM 4 — APPLIED

Maya is saving money. She already has $10 and saves $5 each week. Her brother Jake has $25 and saves $2 each week. After how many weeks will they have the same amount? How much will that be?

PROBLEM 5 — CRITICAL THINKING

Consider y = 4x + 2 and y = 4x − 3. Without solving, explain how many solutions this system has and why. Then change one number in the second equation to create a system with exactly one solution.

SUMMARY

Lesson Summary

A system of linear equations is a set of two (or more) equations that share the same variables. To solve the system, you find the (x, y) pair that makes every equation true. On a graph, this pair is the intersection point — the spot where the lines cross.

You can solve a simple system by graphing both lines and reading the crossing point, or by using substitution to set the equations equal and solve with algebra. Two lines with different slopes always have exactly one solution. Parallel lines have no solution, and identical lines have infinitely many. Always check your answer by plugging it into both equations.