# Solving Linear Equations

#### Common Questions

#### What is an equation?

Great question! Equations use equal signs to tell us that two expressions are **equal**. An example of an equation is $2x+6=12$. If you think something might be an equation, look for the equal sign!

Read below to learn more.

#### Common Questions

Great question! Equations use equal signs to tell us that two expressions are Read below to learn more. | |

Good question! A variable is a symbol (usually a letter) that represents a number we don’t know yet but want to figure out. Read below to learn more. | |

To solve an equation, you'll have to use inverse operations. Inverse operations undo each other! Think about addition and subtraction, or multiplication and division. If you add a number and subtract that same number, they cancel each other out. Read below to learn more. |

Linear equations and variables can seem confusing - but never fear! We're here to help.

### What is an Equation?

An equation uses an equals sign to tell us two values or expressions are equal. Let's say you have an equation like this:

$3x+10=16$The first thing we should notice is this equation mostly just has numbers, but we do have an $x$ in there. This $x$ is what we call a variable. A variable is a symbol, usually a letter like $x$ or $y$, that represents a number we don't know yet but want to find.

So this equation essentially tells us that the value of $x$ can be found by figuring out what number multiplied by $3$ then added to $10$ is equal to $16$. We could plug in a bunch of numbers until we get a match, but that could take a while 😅.

The best way to solve an equation is to isolate the variable, which is the process of "undoing" everything happening to the variable in reverse order.

### Inverse Operations

To undo an equation, we will need to use inverse operations. An inverse operation is an operation that reverses the effects of another operation. For example, if we add a number then subtract the same number, our value won't change. Addition and subtraction cancel each other out and so do multiplication and division.

#### Finding the Foot! 🦶🏿🦶🏾🦶🏽🦶🏼🦶🏻

In the example $3x+10$, we would first multiply $x$ by $3$ then add $10.$

Think of these operations on $x$ as putting shoes & socks on your feet and the inverse operations as taking them off.

When you put them on, you first put on your socks and then your shoes. When you take them off, you reverse the order, taking off your shoes first and then your socks.

Similarly, when performing inverse operations, we want to reverse the order of operations. Last one on = first one off!

Let's try using inverse operations!

Which of the following shows the inverse operations of $3x+10$ in the correct order?

Nice! Now let's put these inverse operations to work and get footloose!

- $3x+10−10→3x$ $(👟🧦🦶🏿)$
- $33x →x$ $(👟🧦🦶🏿)$

Nice! We end up with $x$ all by itself, which means our inverse operations worked!

Now that we know what we should do to isolate $x$, there's just one more thing for us to keep in mind while solving equations.

### Balance!

Think of an equation like a scale or a see-saw. In order to keep things in balance, you want the expressions on the left and right side of the equals sign to be equal to each other. That means if you change something on one side, you have to make the same change on the other. Try playing around with the see-saw below!

### Balanced!

Try moving some of these ⬆️ onto each side and see what happens.

In the previous section, we identified that the correct order of inverse operations for $3x+10$ is subtracting $10$ and then dividing by $3$. Let’s use this to solve the equation $3x+10=16$.

First, we’ll subtract $10$ from both sides:

$3x+10−10=16−10$$3x=6$Then, we'll divide both sides by $3$:

$33x =36 $$x=2$We end up with $x=2$ as our answer! This tells us that $2$ is the value of $x$ that makes this equation true. (If you want to check your answer, you can always plug in $2$ for $x$ in the original equation and see if it’s true.)

Although you may not realize it, we actually solve equations everyday. Let’s take a closer look at some situations involving equations!

### Solving an Equation

**Example 1: **You started playing Fortnite and need to craft some traps. You have $26$ wood planks, and you know that a legendary trap requires $8$ planks while a rare trap requires $6$ planks. If you need one legendary trap, how many rare traps can you craft?

For this problem, our unknown is the number of rare traps you can craft, so we’ll use the variable $x$ to represent that unknown. Given the number planks you have and the number of planks that are needed for each type of trap, which equation represents our scenario?

Awesome job! Since each rare trap requires $6$ planks, you will need $6x$ planks. Adding the number of planks needed to craft the legendary trap gives us a total of $6x+8$. Since we want this to be equal to the amount of planks you own, our equation will be $6x+8=26$.

Now let's use inverse operations to isolate $x$!

If we want to solve $6x+8=26$, what move should we make first?

Superb! Remember we need to do the same thing to both sides to keep the equation balanced, so we’ll subtract $8$ from the right side as well.

$6x+8=26$$6x+8−8=26−8$$6x=18$What move should we make next to isolate $x$?

Awesome job!

$6x=18$$66x =618 $$x=3$We’ll end up with $x$ equaling $3$. In our original equation, $x$ represented how many rare traps we could build, so the solution to this equation tells you that you can craft **$3$ rare traps**.

Let's look at another example!

**Example 2: **For your class’s holiday party, you decide to make sopapillas. If you want to save $15$ sopapillas for your family and divide the remaining ones evenly between your $20$ classmates so that each person gets $2$ sopapillas, how many sopapillas will you need to make?

Let's use $x$ to represent the number of sopapillas you need. After saving $15$ for your family, how many sopapillas will you have left?

Awesome! That's how many sopapillas you'll have left to divide between your classmates. If you have $20$ classmates, which expression represents how many each classmate gets?

That's right! The expression $20(x−15) $ represents how many sopapillas each of your classmates will get. Because you want them to have $2$ each, we will set that expression equal to $2$ to get our equation $20(x−15) $. We can read this equation as "$x$ minus $15$ all divided by $20$ is equal to $2$".

Now let's use inverse operations to solve for $x$!

Which inverse operation should we do first to solve the equation $20(x−15) =2$?

Amazing job! Because we divide by $20$ after subtracting $15$, we want to get rid of the $20$ first:

$20(x−15) =2$$20(x−15) ×20=2×20$$x−15=40$What move should we make next to isolate $x$?

Great work! Adding $15$ to both sides will help us get rid of that $−15$ term:

$x−15=40$$x−15+15=40+15$$x=55$Because $x$ represented the number of sopapillas you needed to make, this answer tells us that you'll have to make $55$ **sopapillas** for your class's holiday party. Wow, that's quite a lot. Better get started!

Now it's your turn. In the Quick Practice below, see if you can spot what went wrong when we tried to solve the equation and how we can fix it!

### Quick Practice 1 / 3

Step 1 | $11x−22−22 =99−22$ $11x=77$ |

Step 2 | $1111x =1177 $ $x=7$ |