Mathematics › 📐 Grade 8 › Systems of Linear Equations: Where Two Lines Collide

Systems of Linear Equations: Where Two Lines Collide

Two equations, two unknowns, one meeting point — find it by graphing, substitution, or elimination.

Grade 8Pre-Algebra / Algebra 1

Systems of Linear Equations: Where Two Lines Collide — illustration

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The big idea: A system of linear equations is just two rules that both have to be true at once. Each equation draws a line, and a single point that satisfies both is exactly where the two lines cross. You can hunt for that point by graphing, or get it exactly with algebra — by substituting one equation into the other or by adding the equations to eliminate a variable.

🎯 By the end, you'll be able to

Explain what the solution to a system of two linear equations represents graphically
Solve a system by substitution
Solve a system by elimination (adding or subtracting equations)
Recognize when a system has no solution or infinitely many solutions

📎 You should already know

Slope and the equation of a line
Solving one- and two-step equations

Two rules, one point

Imagine two cars driving along straight roads. Each road can be described by a linear equation. If the roads cross, there is exactly one point — one pair of coordinates — that lies on both roads. That crossing point is the solution to the system of equations: the one (x, y) pair that makes every equation in the system true at the same time.

A system is just two (or more) equations considered together. Solving it means finding the point where their graphs meet.

🔑 The solution is the intersection point

For a system of two linear equations, the solution is the point (x, y) that lies on both lines — the point where the two lines intersect. Plugging that x and y into either equation must make it true.

\[ \begin{cases} y = m_1x + b_1 \\ y = m_2x + b_2 \end{cases} \]

A system of two linear equations. Solving it means finding the (x, y) that satisfies both lines at once.

Seeing it on a graph

If you graph both lines, the solution is simply the point where they cross. Two lines with different slopes always cross exactly once — so most systems have exactly one solution. Try it below: move each line and watch the crossing point update.

🎮 Systems Collision LIVE

Two lines, one solution: move each line and find where they cross — the solution of the system.

Solving by substitution

Graphing is great for seeing the idea, but reading exact coordinates off a graph is hard when the answer is not a whole number. Substitution solves the system exactly: since both equations equal y, you can replace y in one equation with the expression from the other, leaving a single equation in x.

📝 Worked example: Solve the system: y = x + 1 and y = −x + 5.

Both equations already equal y, so set the right-hand sides equal: x + 1 = −x + 5.
Add x to both sides: 2x + 1 = 5.
Subtract 1: 2x = 4, so x = 2.
Substitute x = 2 back into y = x + 1: y = 2 + 1 = 3.

✓ The solution is <strong>(2, 3)</strong> — the one point that lies on both lines.

Solving by elimination

When neither equation is already solved for y, elimination is often faster: add or subtract the two equations so that one variable cancels out completely, leaving a single equation in the other variable.

\[ \begin{aligned} 2x + y &= 10 \\ x - y &= 2 \\ \hline 3x \phantom{+ y} &= 12 \end{aligned} \]

Adding the two equations makes the y-terms cancel (+y and −y), leaving one equation in x alone.

📝 Worked example: Solve the system: 2x + y = 10 and x − y = 2.

Add the two equations directly: the y and −y cancel, leaving 3x = 12.
Divide by 3: x = 4.
Substitute x = 4 into x − y = 2: 4 − y = 2, so y = 2.
Check in the other equation: 2(4) + 2 = 10. ✓

✓ The solution is <strong>(4, 2)</strong>.

⚠️ Not every system has exactly one solution

If the two lines have the same slope but different intercepts, they are parallel and never meet — the system has no solution. If the two equations describe the same line (same slope and same intercept), every point on the line works — the system has infinitely many solutions.

Check your understanding

1. What does the solution to a system of two linear equations represent on a graph?

The solution is the (x, y) pair that lies on both lines at once — exactly where they cross.

2. Solve by substitution: y = 2x and y = x + 3.

Set 2x = x + 3, so x = 3. Then y = 2(3) = 6, giving (3, 6).

3. Solve by elimination: x + y = 7 and x − y = 1.

Adding the equations: 2x = 8, so x = 4. Then 4 + y = 7 gives y = 3, so (4, 3).

4. Two lines in a system have the same slope but different y-intercepts. How many solutions does the system have?

Same slope, different intercept means the lines never cross, so there is no point that satisfies both equations.

5. Two equations turn out to describe the exact same line. How many solutions does the system have?

If every point on the line satisfies both equations, then every one of those infinitely many points is a solution.

✅ Key takeaways

A system of linear equations is two (or more) equations considered together; the solution is the point that satisfies all of them.
Graphically, the solution is the point where the lines intersect.
Substitution replaces one variable using an expression from the other equation, reducing the system to one equation in one unknown.
Elimination adds or subtracts the equations so one variable cancels out.
Parallel lines (same slope, different intercept) give no solution; identical lines give infinitely many solutions.