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What is a Variable?

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Content What is a Variable?

Rules for Using Variables
Variables in Algebra – Example
Variables in Algebra – Guided Practice
Variables in Algebra – Application
Variables in Algebra – Summary
Variables in Algebra – Frequently Asked Questions

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Eugene L.

Basics on the topic What is a Variable?

Understanding Variables in Algebra – Definition

When embarking on an adventure or solving a riddle, the unknown elements can be the most thrilling. In algebra, the unknowns are represented by variables. These are symbols that stand for numbers that are not yet known or can vary. It's like having a box where you can put any number that fits the situation.

Variables are letters or symbols used in mathematics to represent unknown quantities or quantities that can change. They are essential in creating algebraic expressions and equations that model real-world scenarios.

Rules for Using Variables

Variables can represent any number, not just whole numbers.
You can perform the same operations on variables as you can with numbers (addition, subtraction, multiplication, division).
Variables can be combined with other variables and numbers to form expressions and equations.

What does a variable represent in an algebraic expression?

Can you perform mathematical operations on variables?

Can variables have values other than whole numbers?

Variables in Algebra – Example

Let's imagine that Cello and Jessica are planning the length of their hikes each day. They decide that the distance they will hike each day will be represented by the variable $d$ . If they hike $d$ miles each day and plan to hike for $4$ days, the total hiking distance can be expressed as:

$4 \cdot d$

If they decide that " $d$ " is equal to $3$ miles, the total hiking distance for the trip would be:

$4 \cdot 3 = 12$ miles

This simple algebraic expression allows them to adjust their plans by changing the value of $d$ to suit their energy levels each day.

Variables in Algebra – Guided Practice

Let's guide Jessica and Cello as they calculate the amount of trail mix they will need for their trip. They estimate that each camper will need " $m$ " ounces of trail mix per day.

If they plan to bring enough trail mix for $3$ days, how would we express the total amount of trail mix needed?

Now, if we know that each camper needs $6$ ounces of trail mix per day, we can substitute this value into our expression:

Calculate the total amount of trail mix Jessica and Cello will need for $3$ days.

Variables in Algebra – Application

Now it's your turn to help our campers. They want to calculate the total weight of their backpacks, and they know each backpack will carry " $w$ " pounds of gear, plus an additional $2$ pounds for water.

Write an expression for the total weight of one backpack, including the gear and water.

Suppose that the gear weighs $8$ pounds:

Calculate the total weight of the backpack with the gear and water.

Variables in Algebra – Summary

Key Learnings from this Text:

Variables in algebra represent unknown or changeable quantities.
They can be any letter or symbol, and they allow us to create flexible mathematical models.
You can apply arithmetic operations to variables, just as you would with numbers.
Understanding how to use variables is essential for solving algebraic problems and can be applied to real-world scenarios.

Keep practicing, and you'll become a pro at using variables to navigate through the exciting world of algebra. Don't forget to explore other content on our website platform, such as interactive practice problems and videos, to continue your adventure in learning!

Variables in Algebra – Frequently Asked Questions

What is a variable in algebra?

How do you solve for a variable?

Can a variable represent a negative number?

Why are variables important in algebra?

How do you write an algebraic expression using a variable?

Can you have more than one variable in an equation?

Are variables always represented by letters?

How do you choose which letter to use as a variable?

Can variables be used in formulas?

Do variables always have to be solved?

Transcript What is a Variable?

Mathematically variables in equations

Scout master Blanco and two Junior Explorers are camping in the New Mexico desert. The two Junior Explorers, Jessica and Cello, must successfully survive a night in the desert alone in order to earn the most prestigious badge: The Blue Diamond Survival Patch. They’re accompanied by their trusty handbook, which will advise them on how to deal with the desert’s perilous variables. As night falls, the Junior Explorers are left with their handbook, their courage and their wits. Mr. Blanco is planning the bonfire for the next night. He of course, references his survival handbook.

First example

It says that, in order to make an awesome bonfire. The fire needs 5 pieces of wood to start, and each additional piece adds an hour of burning time. It also says that the number of pieces of wood for an ideal campfire is 11. How can we express this mathematically? To help Mr. Blanco build the ideal campfire, we can substitute other information we know into our equation. There's one piece of information missing, which we call a variable. But this would be annoying to write, so we can just use a letter for the variable, i.e. "x", but make sure to remember what it stands for. After deciding how much wood he’ll need for the next night’s bonfire, Mr. Blanco goes off into the night in search of wood.

Second example - Steaks for coyoties

Now that the Explorers are all by their lonesome, the trepidation sets in. What's this? Is that a coyote? Cello's in luck? To help him in his quest for the Blue Diamond Survival Patch, Cello can use his handbook and the steaks his mom packed for the trip. Cello looks in his handbook and it says that coyotes need one steak to keep them occupied long enough so you can escape. The handbook suggests to set up an equation with the help of variables. The total number of steaks minus the number of coyotes you can distract equals the steaks available for the campers to eat. We know the total number of steaks, 15 the campers and Mr. Blanco only need 5 steaks. The total number of distracted coyotes is our final unknown, which we'll call 'c', for coyotes. Now that we're down to our final variable, Cello can now solve his coyote problem.

Third example - Rattlesnakes

Meanwhile, while preparing for bed, Jessica is interrupted by a familiar sound coming from outside her tent...it must be a rattlesnake...or two...or….MORE?! The only thing Jessica is sure about is that there are rattlesnakes outside her tent, and she brought 4 sacks for catching snakes. She quickly consults her Junior Explorer handbook and it tells her that the best way to deal with rattlesnakes is with a snake scoop and a burlap sack. She can put up to three snakes in a sack. How many snakes could Jessica bag with the sacks she’s brought? We take the total number of sacks Jessica brought, 4 and multiply by the number of snakes she can put in each sack, 3. Finally, we can name our variable, the number of snakes Jessica can catch, 'S'.

In the morning, and without incident, the two Junior Explorers meet at the center of the camp. When they notice a trail. How to make coyote noises? How to sound like a rattlesnake? The Junior Explorers follow the trail of items and it leads them to Mr. Blanco?!? That trickster! These Junior Explorers really deserve their Blue Diamond Survival Patches!

What is a Variable? exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video What is a Variable?.

Explain what a variable is.
Hints

For example, Paul is $12$ years old. The age of Paul's sister, Anne, is unknown. Together they are $21$ years old.
This leads to the equation $12+a=21$ , where $a$ is the variable representing Anne's age.

Say Paul has two apples and five pears and both of them have the same price, $x$ . If he pays $1.40$ dollars in total, then the corresponding equation is $2x+5x=1.4$ .
Because both summands have the factor $x$ in common, we can factor it to get $2x+5x=(2+5)x=7x$ .

An example of an algebraic equation is $2x-2=12$ .

Solution
A variable is a symbol, often $x$ , $y$ , or any letter used to represent an unknown value. For instance,
- $c$ for coyote
- $s$ for snake
This letter represents a value in equations and inequalities.
Variables can be used in algebraic expressions, together with constants, operators, etc. Using variables can be useful when solving word problems.
Variable can be treated in the same way as numbers, as they represent quantities:
- We can add or subtract them $2x+3x=5x$ or $5s-3s=2s$ .
- We can multiply them $(2x)(3x)=6x^2$ .
- We can also divide them $(6x^2)\div(3x)=2x$ .
Find the right equation.
Hints

Not every variable must be $x$ or $y$ . You can also use the first letter of the unknown value, for instance.

Just look at the equation with the corresponding words.

$\le$ or $\ge$ are used for inequalities.

Solution
Let's have a look at what we know:
- Cello has $15$ steaks in total.
- The number of coyotes is unknown. Let's use $c$ to represent the number of coyotes.
- Because each coyote devours one steak, we have to subtract $c$ from $15$ , $15-c$ .
- $5$ steaks need to be left for the campers to eat. So we must have that $15-c$ equals $5$ in the end.
With this knowledge, we can establish the following equation: $15-c=5$ .
Find the words which can be represented by variables.

Hints

Keep in mind that a variable represents an unknown quantity, which would like to be known.

Perhaps it is helpful to establish the equations which represent each word problem. For example, the first one is given by $(15)(x)=60$ .

Solution

When solving a word problem, the first step is determining the unknown values which would like to be known, then assigning a variable to each of these unknown values.
Together with the known values you can establish the corresponding equation.
Lunch
The unknown value is the number of new members. Each given number can't be a variable at all.
Muffins
In this problem the amount of flour for each muffin is wanted.
Scrambled eggs
You know the number of eggs per portion as well as the total number of eggs. The number of portions is wanted.
Firewood
Here we know the total number of collected wood. We also know that Mr. Bianca samples 20 pieces. Cello and Jessica collect the same number, which is unknown.
Establish the equation for the given word problems.
Hints
Check the known values for each word problem. There is still one unknown value. This is the variable.

First, try to establish the equation on your own. Just have a look at the following example:
Jessica prepares a certain number of scrambled eggs portions. For one portion of scrambled eggs she uses $2$ eggs. In total she uses $30$ eggs. This leads to the equation $(2)(x)=30$ .

Equations can be written in different ways. Let's have a look at the following example, $a+3=5$ .
This equation can also be written as
- $3+a=5$ using the commutative property,
- $3=5-a$ , or
- $a=5-3$ using opposite operations.
Solution

In each of the given problems there is a wanted value. Each time the corresponding variable is $x$ .
Filling the Bus
Because $52$ people are already sitting on the bus and in total $80$ people can fit on the bus, the number of people needed to fill the bus is given by the equation $52+x=80$ .
How Many Buses
We want to know the number of buses. We assign $x$ to this number, and we get the equation $(40)(x)=80$ .
How Many People
We have to add the number of males and females together to get the total number of people, or in other words, $28+52=x$ .
Apples
Because each of the $28$ people have the same number of apples, i.e. $x$ , and the total number is given by $84$ , then we get the equation $(28)(x)=84$ .
Find the variable.
Hints

A variable is often a letter, like $x$ , $y$ , or any other letter, and is used to represent an unknown quantity.

$+$ or $-$ are operators, while $=$ is a relation.

Solution
You can recognize a variable as it is usually a letter (you can also use symbols like a star or a smiley face, but maybe they aren't always the most convenient to use). Looking at our given equations, we can see that:
- $x$ , $y$ or $a$ , $b$ , $c$ are variables.
- $10$ , $5$ are numbers.
- $+$ or $-$ are operators.
- $=$ is a relation.
So let's examine the following equations:
- $5+\mathbf{x}=11$
Here the variable is given by $x$ .
- $15-\mathbf{c}=5$
The variable $c$ stands for the unknown number of coyotes.
- $(4)(3)=\mathbf{s}$
$s$ is the variable for the unknown number of rattle snakes.
Write the equation.

Hints

Consider the unknown value and assign the variable $x$ to it.

Next collect the known values, i.e. the number of kittens as well as the number of toys for each kitten.

You can establish different equations using the commutative property. For example, $x+4=8$ or $4+x=8$ .

You can also write the equation above as $4=8-x$ or $x=8-4$ .

Solution

Sue likes to know the total number of toys. Let's represent this unknown by $x$ .
Because she buys $2$ toys for each of her new kittens, $x$ is given by multiplying the given numbers
$(3)(2)=x$ .
Sure, you can move the order of multiplication using the commutative property to get
$(2)(3)=x$ .
You can also divide the equation above by $2$ or $3$ to get
$2=x\div 3$ or $3=x\div 2$ .
However, the solution is always the same $x=6$ . Sue, enjoy your toy shopping and your time with the new kittens.