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# 1.5: Using Equations

Difficulty Level: At Grade Created by: CK-12

## Introduction

GORP

Once the groups were divided, the next step involved food distribution. Some of the best foods to hike with are cheese, peanut butter, candy bars, fruit, beef jerky, dried fruit and energy bars. The best hiking food that there is goes by the name GORP which stands for “Good ole’ raisins and peanuts.” Many people like putting other things into GORP, like M & M’s and nuts!

The leaders gave Travis and Henry, two of the boys in Kelly’s group, a scoop, a scale, some gallon plastic bags and 18 pounds of GORP. Travis and Henry have to split up the GORP into bags so that each member of the group has the same amount. They have to be sure, when they split it, to remember to prepare bags for the two leaders as well.

How much GORP should go into each bag?

Travis and Henry grab a piece of paper and a pencil and begin working this out. They think they have it!

Wait a minute! An equation would be very helpful here. This lesson is all about writing and solving equations. You will learn all about equations and then revisit this problem at the end of the lesson.

What You Will Learn

In this lesson you will complete the following tasks.

• Solve single-variable addition and subtraction equations using mental math.
• Solve single-variable multiplication and division equations using mental math.
• Model and solve real-world problems using simple whole number equations.

Teaching Time

I. Solve Single-Variable Addition and Subtraction Equations Using Mental Math

We have been working with algebraic expressions. Remember that algebraic expressions combine numbers, variables and operations together. When given the value of a variable, we can evaluate any expression.

Take the following example: 10r+11\begin{align*}10r + 11\end{align*}.

If r=22\begin{align*}r = 22\end{align*}, we substitute the value of the variable into the expression and evaluate.

Be careful to follow the order of operations.

10r+11=10(22)+11220+11231\begin{align*}&10r + 11 = 10(22) + 11\\ &220 + 11\\ &231\end{align*}

An equation is a mathematical statement that two expressions are equal. The key thing to notice in an equation is that there is an equals sign.

For example, we can say that

15+7=242\begin{align*}15 + 7 = 24 - 2\end{align*}

Since both sides of the equation equal 22, these equations are equivalent.

15+7=22\begin{align*}15 + 7 = 22\end{align*} and 242=22\begin{align*}24 - 2 = 22\end{align*}

A variable equation is an equation that includes an algebraic unknown, or a variable. If you think about this it makes perfect sense. We have an unknown in an equation so we use a variable to represent the unknown quantity. We call this a variable equation.

Take a look at some of these variable equations:

15t126x25=45=x+9=216=3(x7)+1\begin{align*}15t &= 45 \\ 12 &= x + 9\\ 6^x &= 216\\ 25 &= 3(x - 7) + 1\end{align*}

In variable expressions, we used the value of the variable to evaluate.

Variable equations are different because with equations we already know the value of one side of the equation.

When we have a variable equation, we can solve the equation to figure out the value of the variable.

That is what you are going to learn in this lesson. Let’s begin with solving single variable-equations addition and subtraction equations.

Remember, an equation states that two expressions are equal.

When we solve a variable equation, we are finding the value of the variable that makes the equation true. Take a look at the following variable equation.

Example

y+10=15\begin{align*}y + 10 = 15\end{align*}

The equal sign tells us that y+10\begin{align*}y + 10\end{align*} and 15 have the same value.

Therefore, the value of y\begin{align*}y\end{align*} must be a number that, when added to 10, equals 15.

What could be the value of y\begin{align*}y\end{align*}?

We can ask ourselves, “What number plus ten is equal to fifteen?”

You can use mental math to determine that y=5\begin{align*}y = 5\end{align*} because 5+10=15\begin{align*}5 + 10 = 15\end{align*}.

One advantage of working with equations is that you can always check your work.

Think of it like a balance.

This scale is not balanced. When we solve an equation, we want the scale to balance. One side will be equal to the other side.

When you think you know the value of a variable, plug it into the equation.

If your value for the variable is the correct one, both sides will be equal.

The two expressions will balance!

You can work on a subtraction equation in the same way. Let’s look at an example.

Example

x5=10\begin{align*}x-5=10\end{align*}

This is an equation once again. We want to figure out what number, minus five, is equal to ten. That way both sides of the equations will be equal and be balanced. We ask ourselves the question, “What number take away five is equal to 10?”

We can use mental math to figure out that x\begin{align*}x\end{align*} is equal to 15. When working with small numbers, mental math is the quickest way to figuring out the value of the variable.

Now let’s check our answer. To do this we substitute our answer for x\begin{align*}x\end{align*} back into the equation and see if both sides are equal.

15510=10=10\begin{align*}15-5 &= 10\\ 10 &= 10\end{align*}

You can see that if we pick the correct quantity for x\begin{align*}x\end{align*} that the two halves of the scale will be balanced. If we pick the wrong quantity, one that is too big or too small, then the scales will tip.

Solving equations is often called BALANCING EQUATIONS for this very reason!!

1K. Lesson Exercises

Practice using mental math to solve each equation.

1. x+5=12\begin{align*}x+5=12\end{align*}
2. x3=18\begin{align*}x-3=18\end{align*}
3. 6y=4\begin{align*}6 - y = 4\end{align*}

Take a few minutes to check your work with a friend.

II. Solve Single-Variable Multiplication and Division Equations Using Mental Math

We have just demonstrated how to solve single-variable addition and subtraction equations. To solve variable equations using multiplication and division we follow the same procedure. First, we examine the problem and use mental math.

Using mental math requires you to remember your multiplication tables. Remember that division is the opposite of multiplication, so when using small numbers mental math and your times tables will be your best strategy to balancing equations.

Let’s look at an example.

Example

9p=72\begin{align*}9p=72\end{align*}

Here is a multiplication problem. Whenever you see a variable next to a number it means multiplication. Here we need to figure out, “What number times 9 is equal to 72?”

Using mental math and division, you can see that p\begin{align*}p\end{align*} is equal to 8.

Now let’s check our answer. We do this by substituting the value of the variable, 8, back into the original problem.

9(8)72=72=72\begin{align*}9(8) &= 72\\ 72 &= 72\end{align*}

One side of the equation is equal to the other side, so our equation balances! Our work is complete and accurate.

We can apply mental math when solving division problems too. Remember that a fraction bar means division, so when you see one you know that you are going to be dividing.

Example

x3=4\begin{align*}\frac{x}{3}=4\end{align*}

Remember that when the variable is on top of the fraction bar that we are dividing the bottom number into this number. So we ask ourselves, “What number divided by three is equal to four?”

We use mental math, and our knowledge of the times table, to figure this out. Our missing value is 12.

Next, we can check our work. We substitute 12 back into the equation for the variable x\begin{align*}x\end{align*}.

1234=4=4\begin{align*}\frac{12}{3} &= 4\\ 4 &= 4\end{align*}

One side of the equation is equal to the other side, so our equation balances! Our work is complete and accurate.

1L. Lesson Exercises

Practice solving these equations.

1. 5y=20\begin{align*}5y=20\end{align*}
2. 6g=42\begin{align*}6g = 42\end{align*}
3. x7=49\begin{align*}\frac{x}{7}=49\end{align*}

III. Model and Solve Real – World Problems Using Simple Whole Number Equations

We can use all of this information when working with real-life problems. The key is going to be in deciphering whether the equation is an addition equation, a subtraction equation, a multiplication equation or a division equation. Once you figure out the operation in the equation, you can work on naming the variable and solve the equation.

Let’s look at an example.

Example

There are 63 players in the symphony orchestra. 29 of these players are women. How many players in the orchestra are men? Write a variable equation and solve.

In this problem, we are trying to find the number of men in the orchestra, so let’s give this value the variable x\begin{align*}x\end{align*}. The problem tells us the number of women in the orchestra as well as the total number of players. We know that the number of men players plus the number of women players will equal the total number of players, so we can write the equation, x+29=63\begin{align*}x + 29 = 63\end{align*}. Notice that plus is a key word that means addition. Now we can solve for the value of x\begin{align*}x\end{align*}.

Using mental math, and what we know about "fact families", we can look at the problem and know that because a number (x) + 29 = 63, that if we subtract the number 29 from 63 we get 34. We check our answer by adding 34 + 29 to see if they equal 63. They do, so we know that we have found the correct answer.

There are 34 players in the orchestra that are men.

Here is another example.

Example

Alyssa sold $120 in raffle tickets. If each ticket costs$6, how many tickets did she sell? Write a variable equation and solve.

This problem will involve multiplication because the number of tickets times the cost of each ticket will equal the amount of money Alyssa made. Let’s let y\begin{align*}y\end{align*} represent the number of tickets Alyssa sold. The equation is 6y=120\begin{align*}6y = 120\end{align*}.

Using mental math, you should see the relationship between 6 and 120 is similar to the relationship between 6 and 12.

6×26×20y=12=120=20\begin{align*}6 \times 2 &= 12\\ 6 \times 20 &= 120\\ y &= 20\end{align*}

Alyssa sold 20 raffle tickets. This is our answer.

Writing an equation can help you solve all kinds of real life problems. Remember the hikers? Let’s revisit the problem from the beginning of the lesson and see how we can use an equation to work it out.

## Real Life Example Completed

GORP

Once the groups were divided, the next step involved food distribution. Some of the best foods to hike with are cheese, peanut butter, candy bars, fruit, beef jerky, dried fruit and energy bars. The best hiking food that there is goes by the name GORP which stands for “Good ole’ raisins and peanuts.” Many people like putting other things into GORP, like M & M’s and nuts! The leaders have Travis and Henry, two of the boys in Kelly’s group, a scoop, a scale, some gallon plastic bags and 18 pounds of GORP. Travis and Henry have to split up the GORP into bags so that each member of the group has the same amount. They have to be sure, when they split it, to remember to prepare bags for the two leaders as well. How much GORP should go into each bag? Travis and Henry grab a piece of paper and a pencil and begin working this out.

First, let’s work on writing an equation. There are 10 kids per group + 2 leaders = 12 total members of the group. There are 18 lbs of GORP.

12x=18\begin{align*}12x=18\end{align*}

x=\begin{align*}x =\end{align*} the number of pounds of GORP per person

We can use guess and check to solve this problem. One pound is too small-that would only be 12 pounds of GORP and we have 18 pounds. How about 1.5 pounds? To do this, we multiply 12×1.5\begin{align*}12 \times 1.5\end{align*}.

12× 1.5 18\begin{align*}& \quad \ 12\\ &\underline{\times \ 1.5\;}\\ & \quad \ 18\end{align*}

Each person will have 1.5 pounds of GORP in their bag.

## Vocabulary

Here are the vocabulary words that are part of this lesson.

Algebraic Expression
an expression that contains a combination of numbers, variables and operations. It does not have an equals sign.
Equation
a number sentence with two expressions divided by an equal sign. One quantity on one side of the equation equals the quantity on the other side of the equation.
Variable Equation
an equation where a variable is used to represent an unknown quantity.

## Technology Integration

http://www.mathplayground.com/mv_defining_variables.html – This is a video on defining and identifying variables.

Resources

You can read about teen hiking and GORP at a few websites.

http://walking.about.com/cs/snacks/a/recipetrailmix.htm – Here is a recipe for making your own GORP. For allergies, substitute something else in the mix.

http://www.outdoors.org/recreation/twa/index.cfm – Here is some information about some of the information necessary for teens when hiking in a teen adventure program.

## Time to Practice

Directions: Use mental math to solve each addition or subtraction equation.

1. x+2=7\begin{align*}x+2=7\end{align*}

2. y+5=10\begin{align*}y+5=10\end{align*}

3. a+7=20\begin{align*}a+7=20\end{align*}

4. b+8=13\begin{align*}b+8=13\end{align*}

5. z+10=32\begin{align*}z+10=32\end{align*}

6. s+14=26\begin{align*}s+14=26\end{align*}

7. 4+y=8\begin{align*}4+y=8\end{align*}

8. 6+x=21\begin{align*}6 + x = 21\end{align*}

9. 17+a=23\begin{align*}17 + a = 23\end{align*}

10. 18+b=30\begin{align*}18 + b = 30\end{align*}

11. \begin{align*}12 + x = 24\end{align*}

12. \begin{align*}13 + y = 18\end{align*}

13. \begin{align*}15 + a = 22\end{align*}

14. \begin{align*}x + 17 = 24 \end{align*}

15. \begin{align*}y + 3 = 45\end{align*}

16. \begin{align*}x - 4 = 10\end{align*}

17. \begin{align*}y - 8 = 20\end{align*}

18. \begin{align*}5 - y = 2\end{align*}

19. \begin{align*}22 - a = 15\end{align*}

20. \begin{align*}18 - y = 2\end{align*}

Directions: Use mental math to solve each multiplication or division equation.

21. \begin{align*}5x = 25\end{align*}

22. \begin{align*}6x = 48\end{align*}

23. \begin{align*}2y = 18\end{align*}

24. \begin{align*}3y = 21\end{align*}

25. \begin{align*}4a = 16\end{align*}

26. \begin{align*}13b = 26\end{align*}

27. \begin{align*}15a = 30\end{align*}

28. \begin{align*}15x = 45\end{align*}

29. \begin{align*}\frac{x}{2}=3\end{align*}

30. \begin{align*}\frac{x}{4}=5\end{align*}

31. \begin{align*}\frac{x}{3}=11\end{align*}

32. \begin{align*}\frac{x}{5}=12\end{align*}

33. \begin{align*}\frac{x}{7}=8\end{align*}

34. \begin{align*}\frac{x}{8}=9\end{align*}

35. \begin{align*}\frac{x}{3}=12\end{align*}

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