When we use the equals sign (=), we indicate that two expressions are equal in value. This is called an equation. For example, is an equation. By choosing certain procedures, you can go step by step from a given equation to the equation = some number. The number is the solution to the equation.
One of the first procedures used in solving equations has an application in our everyday world. Suppose that we place a -kilogram box on one side of a seesaw and a -kilogram stone on the other side. If the center of the box is the same distance from the balance point as the center of the stone, we would expect the seesaw to balance. The box and the stone do not look the same, but they have the same value in weight. If we add a -kilogram lead weight to the center of weight of each object at the same time, the seesaw should still balance. The results are equal.
There is a similar principle in mathematics. We can state it in words like this.
The Addition Principle
If the same number is added to both sides of an equation, the results on each side are equal in value.
We can restate it in symbols this way.
For real numbers a, b, c if a=b thenat+tc=b+ec
Here is an example.
If
, then
Since we added the same amount to both sides, each side has an equal value.
We can use the addition principle to solve an equation.
EXAMPLE 1 Solve for .
Use the addition principle to add to both sides.
Simplify.
The value of is .
We have just found the solution of the equation. The solution is a value for the variable that makes the equation true. We then say that the value, , in our example, satisfies the equation. We can easily verify that is a solution by substituting this value in the original equation. This step is called checking the solution.
Check. =
≟
= ✔
When the same value appears on both sides of the equals sign, we call the equation an identity. Because the two sides of the equation in our check have the same value, we know that the original equation has been correctly solved. We have found the solution.
When you are trying to solve these types of equations, you notice that you must add a particular number to both sides of the equation. What is the number to choose? Look at the number that is on the same side of the equation with , that is, the number added to . Then think of the number that is opposite in sign. This is called the additive inverse of the number. The additive inverse of is . The additive inverse of is . The number to add to both sides of the equation is precisely this additive inverse.
It does not matter which side of the equation contains the variable. The term may be on the right or left. In the next example the x term will be on the right.
EXAMPLE 2 Solve for .
Add to both sides, since is the additive inverse of . This will eliminate the on the right and isolate .
Simplify.
The value of is .
Check. =
≟ Replace by .
= ✔ Simplify. It checks. The solution is .
Before you add a number to both sides, you should always simplify the equation. The following example shows how combining numbers by addition separately, on both sides of the equation—simplifies the equation.
EXAMPLE 3 Solve for .
Simplify by adding.
Add the value to both sides, since is the additive inverse of .
Simplify. The value of is .
Check. =
≟ Replace by in the original equation.
✔ It checks.
In Example 3 we added to each side. You could subtract from each side and get the same result. In earlier lesson we discussed how subtracting a is the same as adding a negative . Do you see why?
We can determine if a value is the solution to an equation by following the same steps used to check an answer. Substitute the value to be tested for the variable in the original equation. We will obtain an identity if the value is the solution.
EXAMPLE 4 Is the solution to the equation ? If it is not, find the solution.
We substitute for in the equation and see if we obtain an i