In this explainer, we will learn how to solve equations in which the variable is on each side of the equation.
It can happen that an equation involves the equality of two expressions, each of them containing a variable. For instance, consider the equation
What does this equation tell us? We have an unknown number, called , and the equation says that if we take three times this number and add five, we get the same number as when we take four times this number and take away two. We can represent this equation with a bar diagram.
We see that we need to remove the -term from one side of the equation, so that the variable appears only on one side. The most natural way in this example is to take away from each side, as is shown in the next diagram.
The then disappears from the left-hand side (as it is removed), and we have on the right-hand side .
This leaves us with
Now, we know that adding 2 to is , so adding 2 to both sides will lead us to the solution:
This step can be represented on our bar diagram as well.
We see that we have been actually using the additive and multiplicative properties of equalities, which state that if the same amount is added to (or subtracted from) both sides of the equality, or if both sides of the equation are multiplied (or divided) by the same number, then the equality is still true.
The strategy is therefore to get rid of the -term on one side by subtracting it from both sides of the equation.
In the previous example, , we could have chosen to subtract from each side of the equation. This would have led to
which simplifies to
Subtracting 5 from each side gives
Finding from is just finding the opposite of , which is actually equivalent to multiplying by . The opposite of is 7, so
We find, of course, the same result, only that it was slightly longer in the second case.
Example 1: Solving Equations with Variables on Each Side
Find the value of :
We want to solve the equation
For this, we want all the -terms to be on one side of the equation. To achieve this, we need to subtract from each side of the equation one of the -terms. We can choose to subtract from each side the smaller -term, , so that it disappears from the left-hand side, and we find
Now, we simply need to subtract 1 from each side, so that we will have one -term on one side and a number on the other side:
We can check that our answer is correct by plugging this value into both expressions of our equation. We find
We find the same value for both expressions, so we have, indeed, when .
Hence, the value of that makes the equation true is 4.
Example 2: Solving Equations with Variables on Each Side
Solve the equation .
We want to solve the equation
For this, we want all the -terms to be on one side of the equation. To achieve this, we subtract from each side of the equation one of the -terms. We can choose to subtract from each side the smaller one, , so that it disappears from the left-hand side, where it is now. As has a negative coefficient, subtracting it is actually adding . Indeed, we have . We find
which is, once the -terms have been combined on the right-hand side,
To get rid of the 26 on the right-hand side, we subtract it from each side so that we get
We have found the value of , so dividing both sides by 3 will give us the value of :
We can check that our answer is correct by plugging this value into the original equation and checking that the equation holds true:
Example 3: Solving Equations with Variables on Each Side in a Geometry Context
Find the length of .
We want to find the length of . By observing the diagram, we see that , , and are aligned. Therefore, we have
We only know the length of , while the other two lengths are given in a term of the unknown .
By substituting for the lengths of the segments as given on the diagram, we find
Solving for will allow us then to find the length of , which is .
So, let us solve . The -term in the left-hand side () can be removed from its side by subtracting it from both sides of the equation. This gives us
which simplifies to
We need now to remove the from the side with the -term, so for this we add 2 to each side. We find
Dividing both sides by 2 now gives us the value of ; namely,
We can quickly check that the value we have just found for is correct by plugging it into the original equation . Is the equation true when ?
The left-hand side is
and the right-hand side is
We find the same value for both expressions; hence, our result is correct; namely, .
Now, since the length of is cm, we find that
The length of is 42 cm.
Example 4: Solving Equations with Variables on Each Side in a Geometry Context
Given that , use the information in the figure to find the perimeter of triangle .
We want to find the perimeter of the triangle. Perimeter means “the measure (distance) around”(“peri” means around), so it is the distance around the edge of a shape. Looking at the diagram, we see that the sides of the triangle are given in terms of an unknown, .
Therefore, we need to use other information about the triangle to be able to find the value of and then find the perimeter of the triangle. We are given an important piece of information in the question; that is, . A triangle with two equal angles is isosceles. And we know that it implies that the two sides forming the third angle (here ) are equal. That means that , and so
To have all -terms on one side of the equation, we can subtract from both sides so that it is removed from the right-hand side. We get
which simplifies to
Now, we need to add 5 to each side so that the -term is on its own on one side of the equation. We find
Remember to do a quick check of the value found for at this point. When the value of 6 is plugged into and , do we get the same value for both expressions? The answer is yes, we find a value of 25 for both.
Now that we have found the value of that makes an isosceles triangle, we find that and
The perimeter is then the sum of the three sides; that is,
Our answer is that the perimeter of the triangle is 68 cm.
Example 5: Solving Equations with Variables on Each Side in a Real-Life Problem
The startup cost for a restaurant is , and each meal costs $10 for the restaurant to make. If each meal is then sold for $15, after how many meals does the restaurant break even?
Let us call the number of meals that need to be sold for the restaurant to break even. A business breaks even when the production costs are exactly balanced by the sales incomes.
We are going to write first an algebraic expression for the cost to produce meals. We need to translate mathematically “the startup cost of a restaurant is , and each meal costs .” The total cost in dollars for meals is then , which is written as
Then, we write an algebraic expression for the money earned in dollars when meals are sold. Since each meal is sold for , meals are sold for , which is simply written as .
The restaurant breaks even when these two quantities are equal; that is, when
We need to have both -terms on the same side of the equation, so we subtract from both sides so that the -term on the left-hand side is removed. We have
We now divide both sides by 5 to find the value of :
We have found that the restaurant needs to sell 24,000 meals in order to break even.
- The additive and multiplicative properties of equalities state that if the same amount is added to (or subtracted from) both sides of the equality, or if both sides of the equation are multiplied (or divided) by the same number, then the equality is still true.
- These properties are used to solve an equation of the form , so as to get an equation of the form .