Determining Whether an Integer is a Solution to an Equation

Learning Outcomes

  • Determine whether an integer is a solution to an equation

Determine Whether an Integer is a Solution of an Equation

In Solve Equations with the Subtraction and Addition Properties of Equality, 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 [latex]2x - 5=-13\text{:}[/latex]

1. [latex]x=4[/latex]
2. [latex]x=-4[/latex]
3. [latex]x=-9[/latex]

Solution

1. Substitute [latex]4[/latex] for x in the equation to determine if it is true.
[latex]2x--5=--13[/latex]
 Substitute [latex]\color{red}{4}[/latex] for x. [latex]2(\color{red}{4})--5=--13[/latex]
Multiply. [latex]8--5=--13[/latex]
Subtract. [latex]3\not=--13[/latex]

Since [latex]x=4[/latex] does not result in a true equation, [latex]4[/latex] is not a solution to the equation.

2. Substitute [latex]−4[/latex] for x in the equation to determine if it is true.
[latex]2x--5=--13[/latex]
Substitute [latex]\color{red}{--4}[/latex] for x. [latex]2(\color{red}{-4})--5=--13[/latex]
Multiply. [latex]--8--5=--13[/latex]
Subtract. [latex]--13=--13\quad\checkmark[/latex]

Since [latex]x=-4[/latex] results in a true equation, [latex]-4[/latex] is a solution to the equation.

3. Substitute [latex]−9[/latex] for x in the equation to determine if it is true.
[latex]2x--5=--13[/latex]
Substitute [latex]−9[/latex] for x. [latex]2(\color{red}{--9})--5=--13[/latex]
Multiply. [latex]--18--5=--13[/latex]
Subtract. [latex]--23\not=--13[/latex]

Since [latex]x=-9[/latex] does not result in a true equation, [latex]-9[/latex] is not a solution to the equation.

 

try it

In the following video we present more examples of how to verify that an integer is a solution to a linear equation.