Introduction to Modeling: Background You’ll Need 3

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Introduction to Modeling: Background You’ll Need 3

Solve equations by adding

Solve Equations Using the Addition Property of Equality

In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to “undo” the addition in order to isolate the variable.

But suppose we have an equation with a number subtracted from the variable, such as [latex]x - 5=8[/latex]. We want to isolate the variable, so to “undo” the subtraction we will add the number to both sides.

We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.

addition property of equality

For any numbers [latex]a,b[/latex] , and [latex]c[/latex], if [latex]a=b[/latex], then [latex]a+c=b+c[/latex]

Remember the [latex]17\text{-year-old}[/latex] twins, Andy and Bobby? In ten years, Andy’s age will still equal Bobby’s age. They will both be [latex]27[/latex].

[latex]\begin{array}{c}a=b\\ a+10=b+10\end{array}[/latex]

We can add the same number to both sides and still keep the equality.

How To: Solve an equation using the Addition Property of Equality

Use the Addition Property of Equality to isolate the variable.
Simplify the expressions on both sides of the equation.
Check the solution.

Solve: [latex]x - 5=8[/latex].

Solve: [latex]27=a - 16[/latex].

	[latex]x-5=8[/latex]
Add [latex]5[/latex] to both sides.	[latex]x-5\color{red}{+5}\color{black}=8\color{red}{+5}[/latex]
Simplify.	[latex]x=13[/latex]
Now we can check. Let [latex]x=\color{red}{13}[/latex].	[latex]x-5=8[/latex] [latex]\color{red}{13}\color{black}-5=8[/latex] [latex]8=8\quad\checkmark[/latex]

	[latex]27=a-16[/latex]
Add [latex]16[/latex] to each side.	[latex]27\color{red}{+16}\color{black}=a-16\color{red}{+16}[/latex]
Simplify.	[latex]43=a[/latex]
Now we can check. Let [latex]a=\color{red}{43}[/latex].	[latex]27=a-16[/latex] [latex]27=\color{red}{43}\color{black}-16[/latex] [latex]27=27\quad\checkmark[/latex]