8.1 Solve equations using the subtraction and addition properties (Page 2/6)

Prealgebra Page 2 / 6

When you add or subtract the same quantity from both sides of an equation, you still have equality.

We introduced the Subtraction Property of Equality earlier by modeling equations with envelopes and counters. [link] models the equation $x + 3 = 8 .$

An envelope and three yellow counters are shown on the left side. On the right side are eight yellow counters.

The goal is to isolate the variable on one side of the equation. So we ‘took away’ $3$ from both sides of the equation and found the solution $x = 5 .$

Some people picture a balance scale, as in [link] , when they solve equations.

Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is “1 mass on each side equals balanced.” The next scale has two weights on each side and is balanced. Beside it is “2 masses on each side equals balanced.” The bottom scale has one weight on the left and two on the right. The right side is lower than the left. Beside the image is “1 mass on one side and 2 masses on the other equals unbalanced.”

The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.

Let’s review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.

Solve: $x + 11 = −3 .$

Solution

To isolate $x,$ we undo the addition of $11$ by using the Subtraction Property of Equality.


Subtract 11 from each side to "undo" the addition.
Simplify.
Check:
Substitute $x = −14$ .

Since $x = −14$ makes $x + 11 = −3$ a true statement, we know that it is a solution to the equation.


Add 4 to each side to "undo" the subtraction.
Simplify.
Check:
Substitute $m = −1$ .

	The solution to $m - 4 = −5$ is $m = −1$ .


Use the Addition Property of Equality.
Find the LCD to add the fractions on the right.
Simplify
Check:

Subtract.
Simplify.
The solution checks.


Use the Addition Property of Equality.
Add.
Check:
Substitute $a = 8$ .
Simplify.
The solution checks.


Rearrange the terms, using the Commutative Property of Addition.
Combine like terms.
Add 11 to both sides to isolate $x$ .
Simplify.
Check. Substitute $x = 12$ into the original equation.


Distribute on the left.
Use the Commutative Property to rearrange terms.
Combine like terms.
Isolate n using the Addition Property of Equality.
Simplify.
Check. Substitute $n = 9$ into the original equation. The solution checks.

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8.1 Solve equations using the subtraction and addition properties (Page 2/6)

Solution

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Solve equations that need to be simplified

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