Learning Objectives
- Use the addition property of equality
- Solve algebraic equations using the addition property of equality
Solve an algebraic equation using the addition property of equality
First, let’s define some important terminology:
- variables: variables are symbols that stand for an unknown quantity, they are often represented with letters, like x, y, or z.
- coefficient: Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of 3x is 3.
- term: a single number, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[/latex] are all terms
- expression: groups of terms connected by addition and subtraction. [latex]2x^2-5[/latex] is an expression
- equation: an equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side. Think of an equal sign as meaning “the same as.” Some examples of equations are [latex]y = mx +b[/latex], [latex]\frac{3}{4}r = v^{3} - r[/latex], and [latex]2(6-d) + f(3 +k) = \frac{1}{4}d[/latex]
The following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation [latex]2x-3^2=10x[/latex], the variable is [latex]x[/latex], a coefficient is [latex]10[/latex], a term is [latex]10x[/latex], an expression is [latex]2x-3^2[/latex].
Equation made of coefficients, variables, terms and expressions.
Using the Addition Property of Equality
An important property of equations is one that states that you can add the same quantity to both sides of an equation and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation “balanced.” If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.
Let’s look at a simple numeric equation, [latex]3+7=10[/latex], to explore the idea of an equation as being balanced.
The expressions on each side of the equal sign are equal, so you can add the same value to each side and maintain the equality. Let’s see what happens when 5 is added to each side.
[latex]3+7+5=10+5[/latex]
Since each expression is equal to 15, you can see that adding 5 to each side of the original equation resulted in a true equation. The equation is still “balanced.”
On the other hand, let’s look at what would happen if you added 5 to only one side of the equation.
[latex]\begin{array}{r}3+7=10\\3+7+5=10\\15\neq 10\end{array}[/latex]
Adding 5 to only one side of the equation resulted in an equation that is false. The equation is no longer “balanced,” and it is no longer a true equation!
Addition Property of Equality
For all real numbers a, b, and c: If [latex]a=b[/latex], then [latex]a+c=b+c[/latex].
If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.
Solve algebraic equations using the addition property of equality
When you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.
When the equation involves addition or subtraction, use the inverse operation to “undo” the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to 0, the additive identity.
In the following simulation, you can adjust the quantity being added or subtracted to each side of an equation to see how important it is to perform the same operation on both sides of an equation when you are solving.
Examples
Solve [latex]x-6=8[/latex].
Solve [latex]x+5=27[/latex].
In the following video two examples of using the addition property of equality are shown.
Since subtraction can be written as addition (adding the opposite), the addition property of equality can be used for subtraction as well. So just as you can add the same value to each side of an equation without changing the meaning of the equation, you can subtract the same value from each side of an equation.
Examples
Solve [latex]x+10=-65[/latex]. Check your solution.
Solve [latex]x-4=-32[/latex]. Check your solution.
It is always a good idea to check your answer whether you are requested to or not.
The following video presents two examples of using the addition property of equality when there are negative integers in the equation.
Think About It
Can you determine what you would do differently if you were asked to solve equations like these?
a) Solve [latex]{12.5}+{ t }= {-7.5}[/latex].
What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with decimals.
b) Solve [latex]\frac{1}{4} + y = 3[/latex]. What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with a fraction.
In the following video, two examples of using the addition property of equality with decimal numbers are shown.
The next video shows how to use the addition property of equality to solve equations with fractions.
The examples above are sometimes called one-step equations because they require only one step to solve. In these examples, you either added or subtracted a constant from both sides of the equation to isolate the variable and solve the equation.
With any equation, you can check your solution by substituting the value for the variable in the original equation. In other words, you evaluate the original equation using your solution. If you get a true statement, then your solution is correct.