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Two-Step Equations with Subtraction and Division

Two step equations x/a - b = c

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Solve Equations Involving Inverse Properties of Subtraction and Division

Let’s Think About It

Credit: Richard Leeming
Source: https://www.flickr.com/photos/dickdotcom/560925264/in/photolist-RyTxf-sQecxE-7rQxCp-baAkfZ-wkkpb-7F9qzq-7koF8a-6wSDsc-9jwrP1-i1PMS6-9ji1Fx-7oapJP-c9Dpzm-i62Ekq-q6Q5uH-bhxkjZ-b2ZTXk-7kAb2o-dySBYN-497CtK-b3tjmx-5MxfPx-6wSDpT-izKGTN-6wSDht-6wSDbP-iVmhbY-3VBfn-5M1GFR-ixactg-iHLwax-8bFoci-8bJFcf-7JxN4X-bhxejX-5L2WFs-8hUbXU-igdAHb-87Mkd-5Nfp9Y-e8LSng-8WLeHT-7woThM-979zMM-97cFmb-kFxMRp-8bFopZ-8bFooT-7NdMtw-vvxmS
License: CC BY-NC 3.0

Brandon and Felicia sold rolls of wrapping paper for a school fundraiser. Brandon sold 3 less than half the number of rolls that Felicia sold. Brandon sold a total of 9 rolls of wrapping paper.

Write an algebraic equation to represent \begin{align*}w\end{align*}, Remember that \begin{align*}w\end{align*} is the number of rolls of wrap the number of rolls of wrapping paper that Felicia sold. Then, find the number of rolls of wrapping paper that Felicia sold.

In this concept, you will solve equations involving inverse properties of subtraction and division.

Guidance

To solve a two-step equation, you will need to use more than one inverse operation. You begin solving two-step equations by isolating the variable.

For example, solve for \begin{align*}z\end{align*}.

\begin{align*}\frac{z}{6} - 7 = 3\end{align*}

Your first step should be to use inverse operations to get the term that includes a variable, \begin{align*}\frac{z}{6}\end{align*}, by itself on one side of the equal sign. In the equation, 7 is subtracted from \begin{align*}\frac{z}{6}\end{align*}. So, you can use the inverse of subtraction—addition. Therefore, add 7 to both sides of the equation.

\begin{align*}\begin{array}{rcl} \frac{z}{6} -7 & = & 3\\ \frac{z}{6} -7 + 7 & = & 3+7\\ \frac{z}{6} & = & 10 \end{array}\end{align*}

Next, you can use inverse operations to isolate \begin{align*}z\end{align*}. Since \begin{align*}z\end{align*} is divided by 6, you can use the inverse of division – multiplication.

\begin{align*}\begin{array}{rcl} \frac{z}{6} & = & 10\\ \frac{z}{6} \times 6 & = & 10 \times 6\\ z & = & 60 \end{array}\end{align*}

The answer is 60.

Let’s review your steps to solving this two-step equation.

Credit: CK-12 Foundation
License: CC BY-NC 3.0

Guided Practice

\begin{align*}\frac{x}{6}-9=8\end{align*} 

First, use the inverse of subtraction—addition to isolate the variable.

\begin{align*}\begin{array}{rcl} \frac{x}{6} -9 & = & 8\\ \frac{x}{6} -9 + 9 & = & 8+9\\ \frac{x}{6} & = & 17 \end{array}\end{align*}

Next, you can use inverse operations to isolate \begin{align*}x\end{align*}. Since \begin{align*}x\end{align*} is divided by 6, you can use the inverse of division – multiplication.

\begin{align*}\begin{array}{rcl} \frac{x}{6} & = & 17\\ \frac{x}{6} \times 6 & = & 17 \times 6\\ x & = & 102 \end{array}\end{align*}

The answer is 102.

Examples

Example 1

\begin{align*}\frac{x}{3}-8=9\end{align*}

First, use the inverse of subtraction—addition to isolate the variable.

\begin{align*}\begin{array}{rcl} \frac{x}{3} -8 & = & 9\\ \frac{x}{3} -8 + 8 & = & 9+8\\ \frac{x}{3} & = & 17 \end{array}\end{align*}

Next, you can use inverse operations to isolate \begin{align*}x\end{align*}. Since \begin{align*}x\end{align*} is divided by 3, you can use the inverse of division – multiplication.

\begin{align*}\begin{array}{rcl} \frac{x}{3} & = & 17\\ \frac{x}{3} \times 3 & = & 17 \times 3\\ x & = & 51 \end{array}\end{align*}

The answer is 51.

Example 2

\begin{align*}\frac{y}{7}-2=13\end{align*}

First, use the inverse of subtraction—addition to isolate the variable.

\begin{align*}\begin{array}{rcl} \frac{y}{7} -2 & = & 13\\ \frac{y}{7} -2 + 2 & = & 13+2\\ \frac{y}{7} & = & 15 \end{array}\end{align*}

Next, you can use inverse operations to isolate \begin{align*}y\end{align*}. Since \begin{align*}y\end{align*} is divided by 73, you can use the inverse of division – multiplication.

\begin{align*}\begin{array}{rcl} \frac{y}{7} & = & 15\\ \frac{y}{7} \times 7 & = & 15 \times 7\\ y & = & 105 \end{array}\end{align*}

The answer is 105.

Example 3

\begin{align*}\frac{a}{7}-2=12\end{align*}

First, use the inverse of subtraction—addition to isolate the variable.

\begin{align*}\begin{array}{rcl} \frac{a}{7} -2 & = & 12\\ \frac{a}{7} -2 + 2 & = & 12+2\\ \frac{a}{7} & = & 14 \end{array}\end{align*}

Next, you can use inverse operations to isolate \begin{align*}a\end{align*}. Since \begin{align*}a\end{align*} is divided by 7, you can use the inverse of division – multiplication.

\begin{align*}\begin{array}{rcl} \frac{a}{7} & = & 14\\ \frac{a}{7} \times 7 & = & 14 \times 7\\ a & = & 98 \end{array}\end{align*}

The answer is 98.

Follow Up

Credit: m01229
Source: https://www.flickr.com/photos/39908901@N06/11386379534/in/photolist-imbaWw-5LZtNu-aE6tjK-HrpBB-izL7kj-4gyeSL-imbgcG-scKGU-dBCC2j-ritVhK-dvcL2y-jnL4eH-uboASW-ucpa6Q-7tLDtG-5NfoQj-btwBNu-7xvK1k-6E3ThJ-iHT3S3-7ngvNx-i5h5W6-5KW1xn-izKtdH-2bz6j-9zZqt8-aEajSy-aE6tdr-aE6tgT-pkQr2-7oVMQB-7tEHxy-7tAKLk-7AyoQq-uDSuF-v2AHn-v2ALb-45UpUA-6wWPHN-dE5hir-9zZtcz-9Gn3ft-9GpQCw-9zZj5c-9zZnM4-9GpTw5-9GpVzG-quL2jJ-5FAVyT-dsNyA7
License: CC BY-NC 3.0

Remember Brandon and Felicia’s fundraising?

Brandon sold 3 less than half of Felicia’s total, which was 9 rolls of wrap.

First, use the key words from the question to help you translate the problem into an equation. Remember that \begin{align*}w\end{align*} is the number of rolls of wrap.

\begin{align*}\begin{array}{rcl} && \text{Brand on sold 3 less than half the number that Felicia sold.}\\ && \qquad \qquad \qquad \quad \downarrow \qquad \qquad \downarrow\\ && \qquad \qquad \qquad \ -3 \qquad \quad \ \ \frac{w}{2} \end{array}\end{align*}

So the equation is \begin{align*}\frac{w}{2}-3=9\end{align*}.

Next, use the inverse of subtraction—addition to isolate the variable.

\begin{align*}\begin{array}{rcl} \frac{w}{2} -3 & = & 9\\ \frac{w}{2} -3 + 3 & = & 9+3\\ \frac{w}{2} & = & 12 \end{array}\end{align*}

Then, you can use inverse operations to isolate \begin{align*}w\end{align*}. Since \begin{align*}w\end{align*} is divided by 2, you can use the inverse of division – multiplication.

\begin{align*}\begin{array}{rcl} \frac{w}{2} & = & 12\\ \frac{w}{2} \times 2 & = & 12 \times 2\\ w & = & 24 \end{array}\end{align*}

The answer is 24.

Felicia sold 24 rolls of wrapping paper.

Video Review

https://www.youtube.com/watch?v=9ITsXICV2u0

Explore More

Solve each two-step equation that has division and subtraction in it.

  1. \begin{align*}\frac{x}{5} - 4 = 8\end{align*}
  2. \begin{align*}\frac{y}{6} - 3 = 8\end{align*}
  3. \begin{align*}\frac{x}{7} - 7 = 10\end{align*}
  4. \begin{align*}\frac{x}{8} - 4 = 12\end{align*}
  5. \begin{align*}\frac{y}{7} - 5 = 11\end{align*}
  6. \begin{align*}\frac{x}{4} - 10 = 12\end{align*}
  7. \begin{align*}\frac{y}{4} - 8 = 2\end{align*}
  8. \begin{align*}\frac{x}{3} - 12 = 9\end{align*}
  9. \begin{align*}\frac{a}{5} - 3 = 11\end{align*}
  10. \begin{align*}\frac{b}{4} - 1 = 15\end{align*}
  11. \begin{align*}\frac{x}{2} - 8 = 4\end{align*}
  12. \begin{align*}\frac{a}{7} - 4 = 9\end{align*}
  13. \begin{align*}\frac{b}{4} - 7 = 3\end{align*}
  14. \begin{align*}\frac{x}{8} - 1 = 12\end{align*}
  15. \begin{align*}\frac{y}{6} - 8 = 5\end{align*}
  16. \begin{align*}\frac{x}{2} - 15 = 12\end{align*}

Vocabulary

Algebraic Equation

Algebraic Equation

An algebraic equation contains numbers, variables, operations, and an equals sign.
Equation

Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
inverse

inverse

Inverse functions are functions that 'undo' each other. Formally: f(x) and g(x) are inverse functions if f(g(x)) = g(f(x)) = x.
One-Step Equation

One-Step Equation

A one-step equation is an algebraic equation with one operation in it that requires one step to solve.
Two-Step Equation

Two-Step Equation

A two-step equation is an algebraic equation with two operations in it that requires two steps to solve.
Variable

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

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