Systems of Equations — Core Skills

Systems of Equations: The Basics

Part 1 of 2 — The Basics

A system is two equations that are true at the same time. They share the same $x$ and the same $y$.

The solution is the one pair of numbers that makes both equations true. It is written as a point, $(x, y)$.

Method 1: Substitution

Use this when one equation already tells you what a variable equals.

Worked example. Solve:

$y = 2x$

$x + y = 9$

Step 1. The first equation says $y$ is the same as $2x$. So in the second equation, replace $y$ with $2x$:

$x + 2x = 9$

Step 2. Combine $x$ and $2x$ to get $3x$:

$3x = 9$

Step 3. Divide both sides by $3$: $x = 3$.

Step 4. Put $x = 3$ back into $y = 2x$: $y = 2(3) = 6$.

The solution is $(3, 6)$.

Method 2: Add the two equations

Use this when one equation has $+y$ and the other has $-y$. Stack them and add straight down. The $y$ terms cancel out.

Worked example. Solve:

$x + y = 10$

$x - y = 4$

Add the left sides and add the right sides:

$(x + y) + (x - y) = 10 + 4$

The $+y$ and $-y$ cancel, leaving $2x = 14$, so $x = 7$.

Then put $x = 7$ into $x + y = 10$: $7 + y = 10$, so .

The solution is $(7, 3)$.

Systems of Equations: Practice

Part 2 of 2 — Practice

The steps, every time

1. Look at the two equations. Does one of them already say what a variable equals, like $y = 4$ or $y = 3x$?
2. If yes, put that value into the other equation in place of the variable. Then solve for the letter that is left.
3. If no, check whether one equation has $+y$ and the other has . If so, add the two equations straight down so the terms cancel.