You have reviewed your knowledge of substitution and inverse operations. Both of these ideas are important when solving equations. Recall that when you solve an algebraic equation, the ultimate goal is to use an inverse operation so that the variable is all by itself on one side of the equal sign, and a number is on the other side. You can check your work using substitution.
Some equations involve integers. The integers are the number set that includes all the whole numbers, their opposites, and the number \( 0 \). The numbers \( - 42,\ 18,\ 12, - 5, \) and \( 100 \) are all integers.
For Example
Solve the equation \( 12 = x + 19 \) for \( x \).
The numbers 12 and 19 are both integers. The value of \( x \) will also be an integer. The first step to solving an equation is to identify the operation with the variable and its inverse. Once you know this information, you can solve the equation using the techniques you have already learned.
Click each section below to review how to solve equations with integers.
Identify the inverse of the operation that is with the variable.
The operation with the variable in \( 12 = x\bbox[yellow]{+} 19 \) is addition.
Its inverse is subtraction.
Set up the inverse operation on both sides of the equation.
To solve, you need to subtract 19 from both sides of the equation.
\( \displaystyle 12 = x + 19 \)
\( \displaystyle 12\bbox[yellow]{- 19} = x + 19\bbox[yellow]{- 19} \)
Simplify both sides of the equation.
Simplify the left-hand side of the equation:
\( \displaystyle 12 = x + 19 \)
\( \displaystyle 12\bbox[yellow]{- 19} = x + 19\bbox[yellow]{- 19} \)
\( \displaystyle 12\bbox[yellow]{+ \left( - 19 \right)} = x + 19\bbox[yellow]{- 19} \)
\( \displaystyle - 7 = x + 19\bbox[yellow]{- 19} \)
Simplify the right-hand side of the equation:
\( \displaystyle - 7 = x + 19\bbox[yellow]{- 19} \)
\( \displaystyle - 7 = x + 0 \)
\( \displaystyle - 7 = x \)
Check the solution using substitution.
Substitute \( - 7 = x \) into the original equation. Then simplify.
\( \displaystyle 12 = \left( - 7 \right) + 19 \)
\( 12 = 12 \)
After you verify that your solution is correct using substitution, you can plot it on a number line if requested. Plotting the solution on a number line helps you understand where the solution is located relative to the other numbers in the equation.
Give it a shot!
You can demonstrate your knowledge of solving equations involving integers with the activity below. Respond to each question using complete sentences. Then, at the end of the activity, you can compare your answers to the sample answers.
1. Willem writes the equation \( g - \left( - 5 \right) = 34 \) to model the amount of his allowance that he adds to his change jar this week. What inverse operation should you use to solve the equation?
2. Explain how to solve the equation \( - 2g = - 14 \). State the inverse operation you would use and how you would apply it.
3. The number line below shows the solution of \( 12 = x + 19 \). Compare the value of the solution to the other values in the equation.
A horizontal number line from -8 to 2. The value -7 is plotted.
| Your Responses | Sample Answers |
|---|---|
To solve the equation \( g - \left( - 5 \right) = 34, \) use the inverse operation of subtraction. Simplify the given equation to \( g + 5 = 34 \). Subtraction is the inverse operation of addition. You can also add \( -5 \) to each side of the equation. |
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To solve this equation, use the inverse operation division. On the left-hand side of the equation, the number \( - 2 \) is attached to the variable \( g \) using multiplication. Division is the inverse operation of multiplication. Divide both sides of the equation by \( - 2 \). |
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For \( 12 = x + 19 \), the value of \( x \) is less than the other integers in the equation. |
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