Solving For Variables

Recall that a variable is a placeholder, usually denoted with a letter, which represents an unknown quantity.

Now it is time to learn how to solve for variables. We will examine equations to determine the value of a variable.

The best way to learn is to jump in with some examples!

Variable Constants

A constant is a fixed value. It is "constant." You will often see constants positioned in front of variables. When you see a constant with a variable, it means you must multiply the variable by that constant.

The expression $3x$ is equivalent to $3 \times x$ or $x+x+x$. It literally means three x's.

A single variable by itself, with no specified coefficient, has a coefficient of 1.

$x = 1x$

$3x = 1x + 1x + 1x$

Here are some more examples, for your reference.

Solving With Algebraic Arithmetic

In the examples shown so far, we solved for variables by asking what number times or plus some other number equals our final answer. This works for simple equations, like $1+x=2$, but for more involved equations, it might be more difficult for you to intuitively determine the correct solution this way.

Consider the following equation: $3x + 8 - 2x + 1 = 12$

Now, it's a bit harder to just look at the equation and find the answer by thinking about it. We're still asking the same question, what number multiplied by three plus eight minus three times itself plus one equals twelve? But it's probably more difficult for you to find the answer just by looking at it.

You're now going to learn how to manipulate equations to solve for variables in a step-by-step manner that involves no guesswork.

Properties of Equality

The properties of equality are fundamental rules that allow us to manipulate equations and maintain their equality.

The Additive Property

The additive property of equality says that if we add or subtract the same number from both sides of an equation, the equations remain equal.

Look at this equation: $5=5$

Now, let's see what happens when we add 1 to each side: $5+1=5+1$

$6=6$

If I add the same number to both sides of the equation, the equations remain equal. It does not matter what the number is, so long as it's equally added on both sides.

$5+10=5+10$

$5+20=5+20$

This demonstrates the additive property of equality.

The Multiplicative Property

The multiplicative property of equality works the same way as the additive property, but applies to multiplication and division instead of addition and subtraction. If we multiply or divide the left side of an equation by a number, and do the same to the right side, both sides remain equal.

Multiple Ways To Solve Equations

When applying multiplicative and additive properties, it does not matter which one you do first, so long as you do the same thing consistently to both sides. Sometimes it may make sense to add first, then multiply. Other times, it may make more sense to multiply first, then add. It depends on the situation. It may be helpful to try solving equations multiple ways.

Consider the following equation: $3y+5=35$

If you start by dividing everything by 3, you're left with this:

$\frac{3y+5}{3}=\frac{35}{3}$

Simplify

$y+\frac{5}{3}=\frac{35}{3}$

Subtract $\frac{5}{3}$ from both sides.

$y=\frac{35}{3}-\frac{5}{3}$

Evaluate/simplify

$y=\frac{30}{3}$

$y=10$

Now, let's solve the equation by subtracting 5 from both sides in the first step, instead of dividing by 3.

$3y+5=35$

$3y+5-5=35-5$

$3y=30$

$3y \div 3 = 30 \div 3$

$y = 10$

In this case, I found it easier to start by subtracting 5 than by dividing by 3. This eliminated the need to deal with complex fractions and made the problem easier to solve.

Combining Like Terms

Now, we'll going to deal with some slightly more advanced examples.

When multiple numbers in an equation have like terms, they may be combined. Like terms mean the variable and the exponent on the variable are the same, but the coefficient/constant may differ.

For example, in this equation $3x+5+2x=7$, the terms $3x$ and $2x$ can be combined, because the variable is the same. With like x-terms combined, it becomes $5x+5=7$

Review the following examples.

Two-Variable Equations

Variables become especially useful when we look at equations with multiple variables. One variable acts as the input, the other as the output. For each input, there's some type of output.

Example: Word Problem: A student is selling cookies for a fundraiser. For each box of cookies sold, $3 is raised for the fundraiser (the profit of each sale).

If the variable $x$ represents the number of boxes of cookies sold, and the variable $y$ represents the total funds raised, the above situation could be represented by the following equation:

$y=3x$

So $y$, the total profit, equals $x$, the number of boxes sold, times $3$, the constant profit made off each box.

Now, we can find the profit raised by selling many boxes of cookies.

If the student sells 18 boxes of cookies, how much profit do they make?

$x=18$

$y=3\times18$

$y=54$

The student would raise $54 if 18 boxes of cookies are sold.

We can use the same equation, $y=3x$ to determine how many boxes of cookies we need to sell to meet a certain goal.

If the student wants to raise $200, how many boxes do they need to sell?

In this case, we know the target profit $y=200$.

Now, we need to solve for $x$ given $y=200$.

$200=3x$

$200\div3=3x\div3$

$200\div3=x$

Since both sides are equal to one another, you can just swap the entire left and right sides of the equation whenever you want.

$x=200\div3$

$x=66\frac{2}{3}$

Since the answer is $66\frac{2}{3}$, we will assume the student cannot sell a fractional box of cookies. They must sell 67 boxes or more to get over their goal of raising $200.

Variables in terms of Variables

We can combine like terms, and solve equations in terms of other variables. These equations don't have one definite solution, but a variety of solutions represented by variables.