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2.1 Solve equations using the subtraction and addition properties

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By the end of this section, you will be able to:
  • Verify a solution of an equation
  • Solve equations using the Subtraction and Addition Properties of Equality
  • Solve equations that require simplification
  • Translate to an equation and solve
  • Translate and solve applications

Before you get started, take this readiness quiz.

  1. Evaluate x + 4 when x = −3 .
    If you missed this problem, review [link] .
  2. Evaluate 15 − y when y = −5 .
    If you missed this problem, review [link] .
  3. Simplify 4 ( 4 n + 1 ) − 15 n .
    If you missed this problem, review [link] .
  4. Translate into algebra “5 is less than x .”
    If you missed this problem, review [link] .

Verify a solution of an equation

Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same – so that we end up with a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!

Solution of an equation

A solution of an equation    is a value of a variable that makes a true statement when substituted into the equation.

To determine whether a number is a solution to an equation.

  1. Substitute the number in for the variable in the equation.
  2. Simplify the expressions on both sides of the equation.
  3. Determine whether the resulting equation is true (the left side is equal to the right side)
    • If it is true, the number is a solution.
    • If it is not true, the number is not a solution.

Determine whether x = 3 2 is a solution of 4 x − 2 = 2 x + 1 .

Solution

Since a solution to an equation is a value of the variable that makes the equation true, begin by substituting the value of the solution for the variable.

.
. .
Multiply. .
Subtract. .

Since x = 3 2 results in a true equation (4 is in fact equal to 4), 3 2 is a solution to the equation 4 x − 2 = 2 x + 1 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Is y = 4 3 a solution of 9 y + 2 = 6 y + 3 ?

no

Got questions? Get instant answers now!

Is y = 7 5 a solution of 5 y + 3 = 10 y − 4 ?

yes

Got questions? Get instant answers now!

Solve equations using the subtraction and addition properties of equality

We are going to use a model to clarify the process of solving an equation. An envelope represents the variable – since its contents are unknown – and each counter represents one. We will set out one envelope and some counters on our workspace, as shown in [link] . Both sides of the workspace have the same number of counters, but some counters are “hidden” in the envelope. Can you tell how many counters are in the envelope?

This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are three circular counters and an envelope containing an unknown number of counters. On the right side are eight counters.
The illustration shows a model of an equation with one variable. On the left side of the workspace is an unknown (envelope) and three counters, while on the right side of the workspace are eight counters.

What are you thinking? What steps are you taking in your mind to figure out how many counters are in the envelope?

Perhaps you are thinking: “I need to remove the 3 counters at the bottom left to get the envelope by itself. The 3 counters on the left can be matched with 3 on the right and so I can take them away from both sides. That leaves five on the right—so there must be 5 counters in the envelope.” See [link] for an illustration of this process.

This figure contains two illustrations of workspaces, divided each into two sides. On the left side of the first workspace there are three counters circled in purple and an envelope containing an unknown number of counters. On the right side are eight counters, three of which are also circled in purple. An arrow to the right of the workspace points to the second workspace. On the left side of the second workspace, there is just an envelope. On the right side are five counters. This workspace is identical to the first workspace, except that the three counters circled in purple have been removed from both sides.
The illustration shows a model for solving an equation with one variable. On both sides of the workspace remove three counters, leaving only the unknown (envelope) and five counters on the right side. The unknown is equal to five counters.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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