Before we begin to solve algebraic equations, we should review what a variable is. A variable, represented with a letter in an equation, is an unknown value that you are trying to find. This unknown value is represented by a letter; it can be any letter, but the most commonly used is x. It could be represented by anything—a box, a big question mark, etc.—so don’t let the letter scare you.
Before we can solve for the unknown value, we need to discuss Properties of Equality. There are several properties of equality that help to solve equations or determine the variable in an equation:
Now that we understand properties of equality, we are one step closer to solving for a variable. To do this, we use inverse operations. Inverse operations are pairs of operations that undo each other, or cancel each other out. Addition and subtraction are a pair of inverse operations, because they undo each other, or cancel each other out.
EXAMPLE
As you can see in the following example, if you start with 3 and add 5, this equals 8. However, if you start with 8 and subtract 5, you are back to your original value of 3. Therefore, adding 5 and subtracting 5 undo each other. Similarly, you can see this is true if you look at 12 minus 8, which equals 4. However, if you start with 4 and add 8, you are back to your original value of 12.Multiplication and division are also inverse operations, because they cancel each other out.
EXAMPLE
We know that 2 multiplied by 7 equals 14, but 14 divided by 7 will bring you back to your original value of 2. Therefore, multiplying by 7 and dividing by 7 undo each other. Similarly, 18 divided by 6 equals 3, but if you start with 3 and multiply by 6, you are back to your original value of 18.Finding the solution or solving most equations involves isolating a variable, or in other words, getting the unknown value by itself. To do this, you want to rearrange your equation so that the variable is by itself on one side of the equation, and everything else is on the other side. To rearrange your equation, you can use the operations that are the inverse of the operations appearing in the equation.
Suppose you want to solve the following equation. To do this, you’ll need to isolate the variable x on the left side of the equation.
Because you are adding 10 to the variable x, you need to use the inverse operation to addition, which is subtraction, to cancel out the plus 10. Therefore, you’re going to subtract 10 on the left side of the equation, which means you also need to subtract 10 on the right side of the equation. On the left side of the equation, the +10 - 10 equals 0. So, the x is isolated by itself of the left side of the equation.
Now you have the variable x on the left and 15 on the right, so your solution is x equals 15.
EXAMPLE
The 7 is added to the x. | |
Subtract 7 from both sides to cancel the 7 and isolate the x on the left side of the equation. | |
Our Solution |
The 5 is negative, or subtracted from x. | |
Add 5 to both sides. | |
Our Solution |
Multiplication and division problems use the same method of isolation using the inverse operation. For example, if the equation has a value multiplied by the variable, you would use the opposite function, division, to isolate the variable. Remember, if there is no operation between the number and the variable, we assume the operation is multiplication.
EXAMPLE
With a multiplication problem, we get rid of the number by dividing on both sides. For example consider the following:Variable is multiplied by 4. | |
Divide both sides by 4 to isolate the x on the left side of the equation. | |
4 divided by 4 equals 1. | |
Our Solution |
EXAMPLE
In division problems, we get rid of the denominator by multiplying on both sides. Consider our next example.Variable is divided by 5. | |
Multiply both sides by 5. | |
On the left side of the equation, we have 5 divided by 5 which equals 1. The x is isolated. | |
Our Solution |
Variable is multiplied by -5. | |
Divide both sides by -5. | |
Our Solution |
Our Equation | |
Because the x is multiplied by 1/3, we need to use the inverse operation of division to cancel it out. This may look complicated so another way to cancel it out is to multiply it by the reciprocal, 3/1. | |
Multiplying by 3/1 on the left side of the equation equals 3/3 times x. Multiplying by 3/1 on the right side equals 12/1. | |
Simplify the equation. | |
Our Solution |
IN CONTEXT
Suppose Jamie scored 18 goals during 9 games of soccer. He wants to know how many goals he scored, on average, per game. You can write an equation to represent the situation.
You can multiply the 9 games of soccer by x, your variable, which represents how many goals he scored per game. Lastly, you know that this will equal the total number of goals he scored, 18.
Because the x is multiplied by 9, you need to divide by 9 on both sides of the equation to isolate the x. This simplifies to be x on the left side of the equation, and 2 on the right side of the equation. Therefore, your solution is x equals 2, which means that Jamie scored, on average, 2 goals per game.
IN CONTEXT
Suppose we were told that Rachel earned $40,000 a year for some number of years and ended up making a total of $280,000 over that time period. For how many years did Rachel work? Can you determine an equation to represent the problem and solve the equation?
Here, we know how much Rachel earns each year, but we are told that this was for "some number of years," which alerts us that the number of years worked would be a variable. We can also determine that if we multiply the number of years worked with her salary each year, we can set that number equal to the total amount earned. Using this equation, we can then solve the problem.
Our Equation x = years Divide both sides by 40,000. Our Solution
IN CONTEXT
Paula is going to visit her grandmother who lives 455 miles away. Assuming Paula never stops and averages 70 mph, how long will it take to get to her grandmother’s?
To solve this problem, you can use the following formula:
If you know two of the values, you can always solve for the unknown third value. Let’s plug in what we know and solve. The distance is 455 miles. The rate is 70 mph. Because we know the distance and rate, our variable for this equation will be:
- Distance = Rate X Time
Now solve for T.
Isolate the variable by doing the inverse operation. Because the variable is currently multiplied by 70, the opposite would be to divide the variable by 70. Our Solution
This is one of many real-life examples of how we use variables to solve problems in everyday life.