Have you ever made GORP to go hiking with?

GORP stands for "Good ole' Raisins and Peanuts." However, many people put other things into GORP and still call it GORP.

Jason is mixing the GORP for his whole group. Since there aren't any allergies, he is having a great time putting everything into the GORP.

Jason adds 3 pounds of peanuts, 5 pounds of M&M's, 3 pounds of cashews, 1 pound of raisins, and a questionable amount of dried fruit. When he weighs the GORP it weighs 22 pounds.

How many pounds of dried fruit was added?

**To figure this out, you will need to know how to write an equation using a variable and use mental math to solve it. You will learn all that you need to know in this Concept.**

### Guidance

Previously we worked on ** 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 problem: \begin{align*}10r + 11\end{align*}.

If \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.

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

**The answer is 231.**

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

Thinking about this, we can say that

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

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

\begin{align*}15 + 7 = 22\end{align*} and \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:

\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.**

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

**That is just 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 the following variable equation.

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

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

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

What could be the value of \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 \begin{align*}y = 5\end{align*} because \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 variable for the variable is the correct one, both sides will be equal.**

**The two expressions will balance!**

**That situation had addition in it, what about subtraction?**

You can work on a subtraction equation in the same way.

\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 fifteen is equal to 10?”

**We can use mental math to figure out that \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 \begin{align*}x\end{align*} back into the equation and see if both sides are equal.

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

**You can see that if we pick the correct quantity for \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!!**

Practice using mental math to solve each equation.

#### Example A

\begin{align*}x+5=12\end{align*}

**Solution: 7**

#### Example B

\begin{align*}x-3=18\end{align*}

**Solution: 21**

#### Example C

\begin{align*}6 - y = 4\end{align*}

**Solution: 2**

Now back to the GORP. Here is the original problem once again.

Have you ever made GORP to go hiking with?

GORP stands for "Good ole' Raisins and Peanuts." However, many people put other things into GORP and still call it GORP.

Jason is mixing the GORP for his whole group. Since there aren't any allergies, he is having a great time putting everything into the GORP.

Jason adds 3 pounds of peanuts, 5 pounds of M&M's, 3 pounds of cashews, 1 pound of raisins, and a questionable amount of dried fruit. When he weighs the GORP it weighs 22 pounds.

How many pounds of dried fruit was added?

To solve this problem, we have to write an equation with a variable.

\begin{align*}3 + 5 + 3 + 1 + x = 22\end{align*}

Here are all the quantities, plus the unknown and it is all equal to 22 pounds.

Now we can add.

\begin{align*}12 + x = 22\end{align*}

You can use mental math to solve this.

**Jason put 10 pounds of dried fruit into the mix of GORP.**

### Vocabulary

- 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.

### Guided Practice

Here is one for you to try on your own.

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.

**Answer**

In this problem, we are trying to find the number of men in the orchestra, so let’s give this value the variable \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 following equation.** \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 \begin{align*}x\end{align*}.

Using mental math, we know that there are more than 30 men because \begin{align*}30 + 29 = 59\end{align*}. We need 4 more than 30 men players to equal 63. So \begin{align*}x = 34\end{align*}.

**There are 34 players in the orchestra that are men.**

### Video Review

- This is a James Sousa video on the basics of solving one -step equations.

- This is a James Sousa video on solving one - step equations with addition and subtraction.

### Practice

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

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

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

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

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

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

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

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

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

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

10. \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*}