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 $y = 3$.

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 $-y$. If so, add the two equations straight down so the $y$ terms cancel.
4. Once you know one variable, put it back into either original equation to find the other one.
5. Check your pair in both equations. The solution has to work in both.

A quick reminder about points

A solution is written $(x, y)$. The first number is always $x$ and the second number is always $y$. Mixing up the order is the most common slip on these questions, so read the point out loud: "$x$ is five, $y$ is two."