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## Solve one - step equations using subtraction.

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The language club is fundraising for a trip to study art and architecture in Paris next summer. They have raised $3,500 from helping the community and$4,800 from various donors. They need a total of 12,300 to subsidize the educational trip. Can you write an equation to solve for how much more they need to raise, and then find out that amount? In this concept, you will learn to solve single variable addition equations. ### Solving Single Variable Addition Equations A variable is used to represent a number, quantity, or expression. For example, in the algebraic equation below, the variable \begin{align*}x\end{align*} represents one possible number. \begin{align*}x + 3 = 5\end{align*} To find out what number \begin{align*}x\end{align*} represents, ask yourself, “What number, when added to 3, equals 5?” \begin{align*}2 + 3 = 5\end{align*}, so \begin{align*}x\end{align*} must be equal to 2. When solving more complex equations, such as \begin{align*}x + 34 = 72\end{align*}, it is important to be more systematic and strategic. To solve an equation you should work to isolate the variable. Isolating the variable means getting the variable by itself on one side of the equal (\begin{align*}=\end{align*}) sign. One way to isolate the variable is to use an inverse operation, that is, the ‘opposite’ operation. For example, addition is the inverse of subtraction, subtraction is the inverse of addition, multiplication is the inverse of division, and division is the inverse of multiplication. To solve an equation in which a variable is added to a number, you can use the inverse of addition––subtraction. To do this you need to use the Subtraction Property of Equality, which states: if \begin{align*}a = b\end{align*}, then \begin{align*}a - c = b - c\end{align*}. This means that if you subtract a number, \begin{align*}c\end{align*}, from one side of an equation, you must subtract that same number, \begin{align*}c\end{align*}, from the other side, too, to keep the values on both sides equal. Let’s look at an example. Solve for \begin{align*}x\end{align*}. \begin{align*}x + 34 = 72\end{align*} Use the subtraction property of equality to subtract 34 from both sides of the equation. This will isolate the variable \begin{align*}x\end{align*}. \begin{align*}\begin{array}{rcl} x + 34 &=& 72 \\ x + 34 - 34 &=& 72 - 34 \\ x + 0 &=& 38 \\ x &=& 38 \end{array}\end{align*}The answer is \begin{align*}x = 38\end{align*}. Here is another example. Solve for \begin{align*}b\end{align*}. \begin{align*}1.5 + b = 3.5\end{align*} In the equation, 1.5 is added to \begin{align*}b\end{align*}. So, use the subtraction property of equality and subtract 1.5 from both sides of the equation to solve for \begin{align*}b\end{align*}. \begin{align*}\begin{array}{rcl} 1.5 + b &=& 3.5 \\ 1.5 - 1.5 + b &=& 3.5 - 1.5 \\ 0 + b &=& 2.0 \\ b &=& 2 \end{array}\end{align*}The answer is \begin{align*}b = 2\end{align*}. ### Examples #### Example 1 Earlier, you were given a problem about the language club, who is going to study art and architecture in Paris. They have raised3,500 from one source and $4,800 from another. However, they need a total of$12,300 for the educational trip. Write an equation to solve for how much more they need to raise.

First, let \begin{align*}x\end{align*} be how much money the students still need to raise.

Next, you need to translate the language into a mathematical equation. Add up all of the funding plus what they still need to raise, \begin{align*}x\end{align*}, and set that equal to the total amount needed.

\begin{align*}3,500+4,800+x=12,300\end{align*}

\begin{align*}8,300+x=12,300\end{align*}

Now, use the subtraction property of equality to subtract 8,300 from both sides of the equation. This isolates the variable \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 8,300-8,300+x &=& 12,300-8,300 \\ x &=& 4,000 \end{array}\end{align*}

The answer is the students still need to raise \$4,000.

#### Example 2

The number of gray tiles in a bag is 4 more than the number of blue tiles in the bag. There are 11 gray tiles in the bag.

Write an equation to represent \begin{align*}b\end{align*}, the number of blue tiles in the bag, and then find the value of \begin{align*}b\end{align*}.

First, translate the language into a mathematical equation. “Is means equals, \begin{align*}b\end{align*} is the number of blue tiles, and there are 11 gray tiles.

\begin{align*}& \text{The} \ \underline{\; \text{number of gray tiles}} \ldots \underline{\; \text{is 4 more than}} \ \text{the } \underline{\; \text{number of blue tiles}} \ldots \\ & \qquad \qquad \ \ \ {\downarrow} \qquad \qquad \qquad \qquad \ {\downarrow} \qquad {\downarrow} \qquad \qquad \qquad \quad \ \ \ {\downarrow} \\ & \qquad \qquad \ \ 11 \qquad \qquad \qquad \quad =4 \quad \ \ + \qquad \qquad \qquad \quad \ b\end{align*}

So the equation is \begin{align*}11 = 4 + b\end{align*}.

Solve the equation to find the number of blue tiles in the bag. Use the subtraction property of equality, and subtract 4 from each side of the equation. This isolates the variable.

\begin{align*}\begin{array}{rcl} 11 &=& 4 + b \\ 11 - 4 &=& 4 - 4 + b \\ 7 &=& 0 + b \\ 7 &=& b \end{array}\end{align*}

The answer is there are 7 blue tiles in the bag.

Solve each addition equation for the missing variable.

#### Example 3

\begin{align*}x + 36 = 90\end{align*}

Use the subtraction property of equality and subtract 36 from both sides of the equation. This isolates the variable \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} x + 36 &=& 90 \\ x + 36 - 36 &=& 90 - 36 \\ x+0 &=& 54 \\ x &=& 54 \end{array}\end{align*}

The answer is \begin{align*} x = 54\end{align*}.

#### Example 4

\begin{align*}x + 27 = 35\end{align*}

Use the subtraction property of equality and subtract 27 from both sides of the equation. This isolates the variable \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} x + 27 &=& 35 \\ x + 27 - 27 &=& 35 - 27 \\ x &=& 8 \end{array}\end{align*}

The answer is \begin{align*}x = 8\end{align*}.

#### Example 5

\begin{align*}y + 1.7 = 6.5\end{align*}

Use the subtraction property of equality and subtract 1.7 from both sides of the equation. This isolates the variable \begin{align*}y\end{align*}.

\begin{align*}\begin{array}{rcl} y + 1.7 &=& 6.5 \\ y + 1.7 - 1.7 &=& 6.5 - 1.7 \\ y &=& 4.8 \end{array}\end{align*}

The answer is \begin{align*}y = 4.8\end{align*}.

### Review

1. \begin{align*}x + 7 = 14\end{align*}
2. \begin{align*}y + 17 = 34\end{align*}
3. \begin{align*}a + 27 = 34\end{align*}
4. \begin{align*}x + 30 = 47\end{align*}
5. \begin{align*}x + 45 = 53\end{align*}
6. \begin{align*}x + 18 = 24\end{align*}
7. \begin{align*}a + 38 = 74\end{align*}
8. \begin{align*}b + 45 = 80\end{align*}
9. \begin{align*}c + 54 = 75\end{align*}
10. \begin{align*}y + 197 = 423\end{align*}
11. \begin{align*}y + 297 = 523\end{align*}
12. \begin{align*}y + 397 = 603\end{align*}
13. \begin{align*}y + 97 = 405\end{align*}
14. \begin{align*}y + 94 = 102\end{align*}
15. \begin{align*}y + 87 = 323\end{align*}

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### Vocabulary Language: English

Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.

Isolate the variable

To isolate the variable means to manipulate an equation so that the indicated variable is by itself on one side of the equals sign.

Subtraction Property of Equality

The subtraction property of equality states that you can subtract the same quantity from both sides of an equation and it will still balance.