Systems of Equations

What is a system?

A system of equations is two or more equations that share variables. A solution is a set of values that makes every equation true at the same time. For two lines in 2D, the solution is the point where they cross.

$\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}$

Three ways to solve: graphing, substitution, elimination.

Method 1: Substitution

Solve one equation for one variable, then plug into the other.

Example. $y = 2x + 1$ and $3x + y = 11$.

Substitute: $3x + (2x + 1) = 11 \implies 5x + 1 = 11 \implies x = 2$.

Back-substitute: $y = 2(2) + 1 = 5$.

Solution: $(2, 5)$.

Method 2: Elimination

Line up the equations and add or subtract to make one variable cancel.

Example. $\begin{cases} 2x + 3y = 12 \\ 2x - y = 4 \end{cases}$

Subtract the second from the first: $4y = 8 \implies y = 2$.

Back-substitute: $2x - 2 = 4 \implies x = 3$.

Solution: $(3, 2)$.

If coefficients don't line up, multiply one equation by a constant first.

Example. $\begin{cases} 3x + 2y = 16 \\ 5x - y = 11 \end{cases}$

Multiply the second equation by 2: $10x - 2y = 22$. Add to the first: $13x = 38$… messy. Instead, multiply the second by 2 to cancel $y$ — wait, that gave us $-2y$. Add the equations: $(3x + 2y) + (10x - 2y) = 16 + 22 \implies 13x = 38 \implies x = 38/13$.

Tip: pick whichever variable cancels cleanly. Sometimes there's no clean choice and you just have to grind.

How many solutions can a system have?

Picture	Solutions
Lines cross at one point	1 (unique)
Lines parallel, never cross	0 (inconsistent)
Same line drawn twice	infinite (dependent)

When solving algebraically:

• A contradiction like $0 = 5$ → no solution.
• An identity like $0 = 0$ → infinitely many solutions.

Key takeaways

• A solution makes every equation true at once.
• Substitution: solve one equation for a variable, plug in.
• Elimination: add or subtract equations to cancel a variable.
• Watch for parallel (no solution) or coincident (infinite) lines.