2.3 Solving equations using the subtraction and addition properties  (Page 4/6)

We will practice translating word sentences into algebraic equations. Some of the sentences will be basic number facts with no variables to solve for. Some sentences will translate into equations with variables. The focus right now is just to translate the words into algebra.

Translate the sentence into an algebraic equation: The sum of $6$ and $9$ is equal to $15.$

Solution

The word is tells us the equal sign goes between 9 and 15.

Locate the “equals” word(s).
Write the = sign.
Translate the words to the left of the equals word into an algebraic expression.
Translate the words to the right of the equals word into an algebraic expression.

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Translate the sentence into an algebraic equation: The product of $8$ and $7$ is $56.$

Solution

The location of the word is tells us that the equal sin goes between 7 and 56.

Locate the “equals” word(s).
Write the = sign.
Translate the words to the left of the equals word into an algebraic expression.
Translate the words to the right of the equals word into an algebraic expression.

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Translate the sentence into an algebraic equation: Twice the difference of $x$ and $3$ gives $18.$

Solution

Locate the “equals” word(s).
Recognize the key words: twice; difference of …. and … .	Twice means two times.
Translate.

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Translate to an equation and solve

Now let’s practice translating sentences into algebraic equations and then solving them. We will solve the equations by using the Subtraction and Addition Properties of Equality.

Translate and solve: Three more than $x$ is equal to $47.$

Solution

	Three more than x is equal to 47.
Translate.
Subtract 3 from both sides of the equation.
Simplify.
We can check. Let $y=44$ .

So $x=44$ is the solution.

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Translate and solve: The difference of $y$ and $14$ is $18.$

Solution

	The difference of y and 14 is 18.
Translate.
Add 14 to both sides.
Simplify.
We can check. Let $y=32$ .

So $y=32$ is the solution.

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Key concepts

• Determine whether a number is a solution to an equation.
  1. Substitute the number for the variable in the equation.
  2. Simplify the expressions on both sides of the equation.
  3. Determine whether the resulting equation is true. If it is true, the number is a solution.
  If it is not true, the number is not a solution.
• Subtraction Property of Equality
  • For any numbers $a$ , $b$ , and $c$ ,

    if	$a=b$
    then	$a-b=b-c$

• Solve an equation using the Subtraction Property of Equality.
  1. Use the Subtraction Property of Equality to isolate the variable.
  2. Simplify the expressions on both sides of the equation.
  3. Check the solution.
• Addition Property of Equality
  • For any numbers $a$ , $b$ , and $c$ ,

    if	$a=b$
    then	$a+b=b+c$

• Solve an equation using the Addition Property of Equality.
  1. Use the Addition Property of Equality to isolate the variable.
  2. Simplify the expressions on both sides of the equation.
  3. Check the solution.