Mathematics » Introducing Integers » Solve Equations Using Integers; The Division Property of Equality

Determining Whether a Number is a Solution of an Equation

Determining Whether a Number is a Solution of an Equation

In Mathematics 102, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

How to 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.

Example

Determine whether each of the following is a solution of \(2x-5=-13\text{:}\)

  1. \(\phantom{\rule{0.2em}{0ex}}x=4\phantom{\rule{0.6em}{0ex}}\)
  2. \(\phantom{\rule{0.2em}{0ex}}x=-4\phantom{\rule{0.6em}{0ex}}\)
  3. \(\phantom{\rule{0.2em}{0ex}}x=-9.\)

Solution

Substitute 4 for x in the equation to determine if it is true. 
 Determining Whether a Number is a Solution of an Equation
Determining Whether a Number is a Solution of an EquationDetermining Whether a Number is a Solution of an Equation
Multiply.Determining Whether a Number is a Solution of an Equation
Subtract.Determining Whether a Number is a Solution of an Equation

Since \(x=4\) does not result in a true equation, \(4\) is not a solution to the equation.

Substitute −4 for x in the equation to determine if it is true.Determining Whether a Number is a Solution of an Equation
Determining Whether a Number is a Solution of an EquationDetermining Whether a Number is a Solution of an Equation
Multiply.Determining Whether a Number is a Solution of an Equation
Subtract.Determining Whether a Number is a Solution of an Equation

Since \(x=-4\) results in a true equation, \(-4\) is a solution to the equation.

Substitute −9 for x in the equation to determine if it is true. 
 Determining Whether a Number is a Solution of an Equation
Substitute −9 for x.Determining Whether a Number is a Solution of an Equation
Multiply.Determining Whether a Number is a Solution of an Equation
Subtract.Determining Whether a Number is a Solution of an Equation

Since \(x=-9\) does not result in a true equation, \(-9\) is not a solution to the equation.

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