Algebraic Expressions and Equations |

**Algebraic Expressions: Saying it with Algebra**

Don and John the quarrelsome twins were at it again. This time, they were fighting about their marbles. It seems both of them collected red ones and both collections had been dumped on the rug and had rolled together. So, they were fighting about how many marbles each one owned. Don said " I'm not sure how many marbles were mine, but I know that **I had eight more than you** did!" Then John suggested they **count all the marbles** on the rug.

" How will that help?" asked Don. "We still won't know how many marbles belong to each one of us. " Well maybe we can figure it out" said John. So, they counted all the marbles and got **44**.

"Now," said John, "we know that there are** 44 marbles in all**, and **you had eight more** than I did. Let's write it down and think about it." So here is what they wrote:

John's marbles + Don's marbles = 44, and

Don's marbles = John's marbles + 8

so, John's marbles + (John's marbles + 8) = 44

Then they had a brilliant idea – at exactly the same time. They decided to **remove 8 marbles** from the group, so that **half of the remaining marbles** were John's, and half were Don's.

If they separated the remaining marbles into two equal parts, they would know how many each one had owned originally.So now they wrote this:

John's marbles + John's marbles = 36 (44 – 8)

or 2 × John's marbles = 36

and from this statement they knew that

**John** must have had **18 marbles** originally,

and **Don**, with 8 more -- had **26**.

What the twins did was solve their mathematical problem with an **algebraic approach**.

Let's write an algebraic expression and equation to show what they did:

Since algebra uses symbols (letters or variables) to represent values, words etc.,

We'll let **J** represent **the number of marbles ****John**** had** originally

So, **J + 8** represents **the number of marbles ****Don**** had** originally

therefore **J**** + (****J + 8****)** represents the **sum total** of both marble collections.

We know the total is 44

So, **J**** + ****J + 8**** = 44** which is the same as **2J + 8 = 44**

Then they removed 8 marbles to get

(2**J + 8) – 8 = (44) – 8 **

And since **8 – 8 = 0**, this statement is the same as **2J = 36**

therefore, the value of **J** must be **½ of 36** which is **18**.

The twins concluded that **John** had **18 marbles** originally,

and **Don**, with 8 more -- had **26**.

Since **18 + 26 = 44**, they knew they had figured it out correctly.

**Algebraic Expressions**

Unless we're *newbies*, when our online chat buddy types **LOL** in response to a joke, we know exactly what it means. It tells us that he or she is "**laughing out loud**". Though we may not be aware of it, we're using algebra to communicate with each other, since we **assign **or** name a recognizable symbol** **to represent something** -- a concept, situation, or a value.

In our solution to the twins' problem, we used **J** to represent the number of marbles John had,

and **J + 8** for the number of marbles Don had. These are **algebraic expressions**.

Then we wrote an **equation** that said the sum of all the marbles is **equal to** 44. From that information, we were able to **solve** for the number of marbles each twin owned.

The first step in creating an algebraic expression is to **define the variable**. This is sometimes done **with a diagram** but most often with a **let statement**. In our solution above, we **let J represent John's marbles**. Once we defined what **J** stood for, we knew **J + 8** represented the number of Don's marbles. And once we **equated** the sum to 44, we could solve the problem.

It's a good idea to choose variables that remind us of the quantity we're describing. We usually set length = *l*, width = *w*, height = *h* etc. Notice how we used ** J** for John's marbles.

.

**Example:
**Say we own a very exclusive toy store that sells only 2 items –

Bears sell for $7 each, and hoops cost $2 each. Now, say a customer buys a few of each item.

If we let

**7***b*** + 2***h*

Remember that in algebra, we indicate multiplication by writing the terms together,

so **6 k** means

Once we know how many of each item our customer wants to buy, we can **evaluate the expression** to find the cost of the purchase (before taxes), since we now have **assigned values** for ** b **and

A customer who buys

Notice how we don't bother to write 7 × 1 – we just write 7.

**An algebraic expression is a SYMBOLIC representation of a situation.**

**Examples:**

1) If *a* represents the price of an apple and *b* represents the price of a banana, write

an algebraic expression for the price of:

a dozen apples 12 |
3 apples and 7 bananas 3 |
5 bananas 5 |
half dozen of each. 6 |

2) **Evaluate** each of the expressions in #1 if the prices are **5¢** per apple and **8¢ **per banana.

a dozen apples 12 |
3 apples and 7 bananas 3 |
5 bananas 5 |
half dozen of each. 6 |

Though we see equal signs in the statements above, they are not really equations. Yes, they're statements of equality but there are no **variables** (letters) in them.

.

3) An algebraic expression for the Perimeter of a rectangle with **base = 5** and **height = h** is

Formulas such as **A = l × w** for the area of a rectangle, are also statements of equality, but, they too are not equations because there's nothing to solve for. We use them as templates or stencils to evaluate certain measures.

Now get a pencil, an eraser and a note book, copy the questions,

do the practice exercise(s),then check your work with the solutions.

If you get stuck, review the examples in the lesson, then try again.

**Practice Exercise 1:** algebraic expressions

1) Write an algebraic expression for the Perimeter and Area of each figure.

2) Evaluate (find a number value for) each algebraic expression in #1 given that:

** b = 7** inches,

**Algebraic Equations: Solving it with Algebra**

this is an equation

**An Equation is an algebraic statement of Equality
between knowns (constants) and unknowns (variables).**

An algebraic equation behaves exactly the same way a seesaw does or balanced scales do, with the equal sign replacing the pivot point of the seesaw or scales. **Whatever we do to one side** of the equation **we do to the other side** in order to **maintain equality or balance**. Those are the operational words. We must **maintain equality at all times** -- why do you think we call it an **EQUATION**?? We would call it something else if it was something else -- but it's not.

In the twins question, the equation was **J**** + ****J + 8**** = 44**

when we removed 8 marbles, we subtracted 8 from both sides of the statement

and got **2J = 36**.

Then we divided by 2 or took ½ of 36 to get 18.

What we did was to **perform the inverse operation** for each one indicated in the equation. We **eliminate** the **additional** 8 marbles when we **subtract** (or remove) 8 from **both sides** of the equation and then, since **we know twice J** but **want just** **one J**, we **divide** both sides of the equation **by 2**.

Now let's solve a problem with an equation from the example about the apples and bananas.

Say we know an **apple costs 5¢** but we **don't know** the price of a banana.

At the check out, we find that **3 apples** and **7 bananas** cost **71¢**.

So, we **let b **represent the

**3(5¢) + 7 b = 71¢** which simplifies to

Now, we'll be brilliant like the twins and take away the 15¢ we pay for the 3 apples.

We get **15 + 7 b – 15 = (71 – 15)¢ **which becomes

This says that the price of **7 bananas** is **56¢** so we know that each banana costs **8¢**, which is **one seventh** of 56. Again, we **solved** this equation **with 2 operations** -- first **subtraction** and then **division** because the operations shown in it were addition and multiplication.

When we solve an equation, our job is to **find** the number **value** of the **unknown **or** variable**. In this example, the **variable **was** b**,

Here's the apples and bananas picture.

If we see subtraction in the equation, we will add. If we see division, we will multiply. The only rule is **no prejudice!! What we do to one side, we do to the other -- ALWAYS!!**

**Examples**

1) Solve this equation: *x* – 7 = 33

We see subtraction, so we will add 7 to both sides to get: (** x – 7) + 7 = 33 + 7**. So

To check our work, we verify if the

2) Solve this equation: **3 x – 4 = 29**

We see subtraction, so we will add 4 to both sides to get:

Which, when we add + 4 and – 4 to get 0 on the left gives us:

Now we see multiplication, so we divide both sides by 3 to get

When we check our answer in the original statement we see it is true. 3(11) – 4 does = 29.

3) Solve this equation:

Here we see addition and division (remember, fractions mean division), so we'll first subtract 9 from both sides and then we will **multiply by 5** to find *x*. We'll get:

This says that when we divide *x* by 5, we get 6, so *x* must be 5 × 6 or 30.

Now get a pencil, an eraser and a note book, copy the questions,

do the practice exercise(s),then check your work with the solutions.

If you get stuck, review the examples in the lesson, then try again.

**Practice Exercise 2:** algebraic equations

1) Solve these equations by addition and division:

a) 5x – 19 = 31 |
b) 2x – 10 = 12 |
c) 7x – 1 = 139 |
d) 4x – 13 = 35 |

2) Solve these equations by subtraction and division:

a) 5x + 19 = 74 |
b) 2x + 10 = 48 |
c) 7x + 17 = 73 |
d) 4x + 13 = 45 |

3) Solve these equations by addition or subtraction and multiplication:

a) | b) | c) | d) |

**hint**: In d) you need addition, multiplication and division, in that order.

4) Harry wants to by new shoes that cost $78. He states that if he had $6 more,

he would have half the price of the shoes.

a) write an equation that states in algebra what the question states in words.

b) Solve the equation to find how much money Harry has right now?

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**Solutions**

**Practice Exercise 1:** algebraic expressions

1) Write an algebraic expression for the Perimeter and Area of each figure.

**Practice Exercise 2:** algebraic equations

1) Solve these equations by addition and division:

a) 5x – 19 = 31 5 |
add 19 |
b) 2x – 10 = 12 2 |
add 10 to both sides |

c) 7x – 1 = 139 7 |
add 1 |
d) 4x – 13 = 35 4 |
add 13 to both sides |

2) Solve these equations by subtraction and division:

a) 5x + 19 = 74 5 |
subtract 19 |
b) 2x + 10 = 48 2 |
subtract 10 |

c) 7x + 17 = 73 7 |
subtract 17 |
d) 4x + 13 = 45 4 |
subtract 13 |

3) Solve these equations by addition or subtraction and multiplication:

a)
so |
add 1 to both sides |
b)
so |
subtract 3 multiply by 7 |

c)
so |
add 4 to both sides multiply both sides by 6 |
d) so 2 |
add 1 to both sides multiply by 3 divide by 2. |

4)

a) Let *a* = the amount of money Harry has, so the equation is:

*a *+ 6 = ½ (78)

b) How much money does Harry have right now?

Since ½(78) = 39, our equation is: *a *+ 6 = 39 -- so we subtract 6 from both sides

to find that Harry has 39 – 6 or **$33** right now.

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