Systems of
Linear Equations

Discover how two equations can share one solution, and learn to find the intersection point where their lines cross.

Pre-Algebra

What Is a System of Equations?

A system of linear equations is a pair of equations with the same two variables, $x$ and $y$. We want to find a single pair $(x,y)$ that makes both equations true simultaneously.

Each equation describes a straight line. The solution is the point where the two lines cross on the coordinate plane.

Where Two Lines Meet

Why Systems Matter

Systems appear whenever two conditions must be satisfied at once:

Shopping

Two friends buy different combos of tacos and drinks for different totals: find each price.

Speed

Two runners start at different spots, run different speeds: when do they meet?

Saving

You save $\$5$/week from $\$20$; a friend saves $\$3$/week from $\$40$: when are you equal?

The Intersection Rule

At the intersection both equations give the same $y$-value for the same $x$-value. That means we can set the two right-hand sides equal and solve for $x$.

Three Possible Outcomes

One Solution

Lines cross at exactly one point. They have different slopes.

No Solution

Lines are parallel: same slope, different intercepts. They never meet.

Infinite Solutions

Lines are identical: same slope and same intercept. Every point is shared.

Graphing vs Substitution

	Graphing	Substitution
How	Plot both lines, read intersection	Set expressions equal, solve
Best when	You want a visual picture	You need an exact answer
Weakness	Hard to read non-integer points	No visual confirmation
Speed	Slower (must draw)	Faster (just algebra)

Solving by Substitution

1

Isolate $y$ in each equation so both read $y = \dots$

2

Set equal: since both equal $y$, set the right sides equal to each other.

3

Solve for $x$: combine like terms, isolate $x$.

4

Find $y$: plug $x$ back into either original equation.

5

Check: substitute $(x,y)$ into both equations to verify.

Example 1: Simple System

1

Both already in $y = \dots$ form. ✓

2

Set equal: $x + 1 = -x + 5$

3

Add $x$ to both sides: $2x + 1 = 5$. Subtract 1: $2x = 4$. Divide: $x = 2$.

4

Plug in: $y = 2 + 1 = 3$. Answer: $(2,3)$.

5

Check: $y=x+1 \rightarrow 3=2+1$ ✓. $y=-x+5 \rightarrow 3=-2+5$ ✓.

Example 2: Steeper Slopes

1

Set equal: $2x - 1 = -x + 5$

2

Add $x$: $3x - 1 = 5$. Add 1: $3x = 6$. Divide: $x = 2$.

3

$y = 2(2) - 1 = 3$. Solution: $(2,3)$.

4

Check: $y=2x-1 \rightarrow 3=2(2)-1$ ✓. $y=-x+5 \rightarrow 3=-2+5$ ✓.

Example 3: Graph + Algebra

Example 4: Fractional Answer

1

$2x + 1 = -x + 3$

2

$3x = 2$, so $x = \frac{2}{3}$.

3

$y = -(\frac{2}{3}) + 3 = \frac{7}{3}$. Solution: $(\frac{2}{3}, \frac{7}{3})$.

4

Check: $\frac{7}{3}=2(\frac{2}{3})+1$ ✓. $\frac{7}{3}=-(\frac{2}{3})+3$ ✓.

Starting from Standard Form

Not every system starts with $y = \dots$. Sometimes you see standard form like $2x + y = 10$. Just isolate $y$ first.

Common Mistakes

✕ Forgetting to Check

Always plug your answer into both equations. If it fails one, the work has an error.

✕ Sign Errors

When moving terms across the equals sign, remember to flip the sign. $-x$ becomes $+x$.

✕ Solving for Only $x$

The solution is a pair $(x, y)$. Do not stop after finding $x$: plug it back to get $y$.

Summary of Methods

Step	What to Do	Example
1. Isolate $y$	Rewrite as $y = mx + b$	$y = x + 1$
2. Set equal	Right sides equal	$x + 1 = -x + 5$
3. Solve $x$	Combine, isolate	$x = 2$
4. Find $y$	Substitute back	$y = 3$
5. Check	Both equations true?	$3=2+1$ ✓, $3=-2+5$ ✓

Question 1 of 5

Solve the system: $y = x + 4$ and $y = -x + 6$. What is the solution?

Setting equal: $x + 4 = -x + 6$ gives $2x = 2$, so $x = 1$. Then $y = 1 + 4 = 5$. The solution is $(1, 5)$.

Question 2 of 5

Two lines have the equations $y = 3x$ and $y = x + 8$. Where do they intersect?

$3x = x + 8$ gives $2x = 8$, so $x = 4$. Then $y = 3(4) = 12$. The intersection is $(4, 12)$.

Question 3 of 5

If two lines have the same slope but different y-intercepts, how many solutions does the system have?

Lines with the same slope but different intercepts are parallel. Parallel lines never intersect, so there is no solution.

Question 4 of 5

Solve: $y = -2x + 10$ and $y = x + 1$.

$-2x + 10 = x + 1$ gives $-3x = -9$, so $x = 3$. Then $y = 3 + 1 = 4$. The solution is $(3, 4)$.

Question 5 of 5

You find that $x = 2$ from a system. You plug it into the first equation and get $y = 7$, but plugging into the second gives $y = 5$. What happened?

A correct $x$ must give the same $y$ in both equations. Getting different $y$-values means there is an algebra error somewhere in the solving process.

Step	What to Do	Example
1. Isolate \(y\)	Rewrite as \(y = mx + b\)	\(y = x + 1\)
2. Set equal	Right sides equal	\(x + 1 = -x + 5\)
3. Solve \(x\)	Combine, isolate	\(x = 2\)
4. Find \(y\)	Substitute back	\(y = 3\)
5. Check	Both equations true?	\(3=2+1\) ✓, \(3=-2+5\) ✓

Systems ofLinear Equations

What Is a System of Equations?

Where Two Lines Meet

Why Systems Matter

The Intersection Rule

Three Possible Outcomes

Graphing vs Substitution

Solving by Substitution

Example 1: Simple System

Example 2: Steeper Slopes

Example 3: Graph + Algebra

Example 4: Fractional Answer

Starting from Standard Form

Common Mistakes

Summary of Methods

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Systems of
Linear Equations