Objective a - Evaluating Variable Expressions

11.1 VARIABLE EXPRESSIONS

Objective A - Evaluating Variable Expressions

In Algebra, it is necessary to talk about a value without knowing what it is.
Variablesare used to represent these values.
Any letter of the alphabet (x, y, z, n)used in this way is called aVariable.

The root word is vary, which means it can change.
Variablevalues change depending on the expression or equation in which it is used.
An expression that contains one or more variables is called a Variable Expression.
Thevariable is a blank that getsreplaced with an actual value.

For Example:

If Ibrought 2 cookies for everyone in this class, the number of cookies I would need would vary depending on how many people showed up for class that day. We can let x equal the number of students who are here for class. Then the number of cookies I would need to bring is:

Since there are only 9 students in class today, how many cookies we would need?

That’s the idea to use in problems within this section.
You’ll be given a variable expression and the value of each variable in the expression.
You “plug in” (mathematical word is substitute) the values for the appropriate variables.

Ex.Evaluate 2x + 4xy when x = 8 and y = 3.

Think of it as +where 8 substitutesthe diamond and 3 substitutesthe circle.

Then you use the Order of Operations Agreement to actually find the value.

Ex.Evaluate whenb = 6 and c = –2.

Ex.Now use the same expression as the previous example, but let b = –12 and c = 3. (Remember: They’re variables, so their values can vary)

Objective B - Simplifying Variable Expressions Containing No Operations in Parentheses

DEFINE:

Terms: The addends of an expression. A term is “held together” with themultiplication operation.
Constant Term: A number with no variable being multiplied by it.
Variable Terms: Terms that have one or more variables as part of the term.

Note: Terms are separated from each other by addition or subtraction, because subtraction is just “addition of the opposite”.

Ex.How many terms are in each of the following expressions?

Expression: / How many terms? / Variable Term(s): / Constant Term(s):

Each variable term has two parts:
The number in front is called the numerical coefficient (or just coefficient);
AND, the variable part (which is everything else).

Ex.For the following terms, determine what the numerical coefficient is and what the variable part is.

Term: / Numerical Coefficient: / Variable Part:

Like Terms: These are terms that have identical variable parts. Determine if the following terms are like terms.

Terms: / Are they like terms?

When you add (and subtract) terms:
Only like terms can be combined together into one term.

All you have to do is add (or subtract) the numerical coefficients out in front, and leave the variable part there as a “tag-a-long.”

Ex. Simplify: Ex. Simplify:

Objective C - Simplifying Expressions Containing an Operation within the Parentheses

Recall: The Commutative and Associative Properties of Multiplication from the handout in Chapter #1.

Simplify the following:

Ex.Ex.

There is a new property to learn now, a very helpful one.

Distributive Property (or the Distributive Property of Multiplication OVER Addition)

If a, b, and c are three numbers:
Then,a(b + c) = ab + ac
Also,a(b – c) = ab – ac

The Distributive Property means that you can “get around” the Order of Operations Agreement.

Ex.Ex.

Use the Distributive Property and then combine like terms, as in the following examples.

Ex.Ex.

11.2INTRODUCTION TO EQUATIONS

Objective A - Determining Whether a Given Value is a Solution of an Equation

An equation is a statement that two expressions are equal.
In other words, there is an equal’s sign.
Up to now, we’ve only worked with algebraic expressions without equal’ssigns.

To determine if a value is a solution to an equation:
Substitute the value into the equation to see if it’s true.
If it’s TRUE – it IS a solution.
If it’s FALSE – it’s NOT a solution.

Ex.Is 2 a solution of 3x(x – 3) = x – 8?

(In the next section, you will have to come up with the actual solution value on your own. Here you are given the solution to check.

Ex.Is –4 a solution of x(x + 4) = x2 + 16?

Objective B - Solving an Equation of the Form x + a = b

Tosolve an equation means that you find the solutionto the equation.
To do this:
Place the variable alone on one side of the equation and a number on the other side:

Variable = Constant.

You will need to use the following two properties.

The Addition Property of Equations

The sum of a term and zero is the term.
Then,a + 0 = a
AND,0 + a = a

The Addition Property of Equations

If a = b, then a + c = b + c
In other words, “You can add the same value to both sides of an equation.”

Ex. Solve: x + 9 = – 7Look at the left-hand side of the equation. What can you add to the left side of the = sign to “get rid of” the 9?

Verify you got the correct answer by checking it using what you did in Objective A to see if it’s a true number statement.

Ex. Solve: y – 6 = 16Ex. Solve: x + 7 = – 8

Major Point:
To “get rid of” an added number, subtract the same number from both sides.
To “get rid of” a subtracted number, add that number to both sides. (i.e. always perform the opposite operation.)

Objective C - Solving an Equation of the Form ax = b

There are two properties you must use to solve these types of equations in order to find:
Variable = Constant.

Multiplication Property of Reciprocals

The product of a nonzero number and its reciprocal is 1.
Then,
Ex.

Multiplication Property of 1

The product of 1 and any term is that same term.
Then,
AND,
Ex.

The Multiplication Property of Equations

If a = b, then (if c 0) ac = bc
In other words, “You can multiply both sides of an equation by the same value.”

Conversely, if a number is multiplied by the variable:
To “get rid of” the number, you need to multiply by its reciprocal,which is the same as dividing by that number.

Ex. Goal is to get the variable by itself. What can you multiply? to get x by itself?

Ex. 4y = -28Ex. –56 = 7x

11.3GENERAL EQUATIONS: PART 1

Objective A - Solving An Equation of the Form ax + b = c

An example of this type of equation is: 4x + 8 = 20

You will need to use what you learned in Section 11.2, both Objectives B and C.
That means you still need to get the equation in the form:

x = number

There are two numbers to “get rid of”→ the 8 and 4.
Which one do you “get rid of” first?

Procedure:

ALWAYS isolate the x-term first (that means to get the x-term by itself on one side of the = sign):
x-term = number
To get the x-term by itself, always add/subtract to “get rid of” the constant term first.
Next, you need to get just the variable by itself:
x = number
To get just the x by itself, always multiply/divide to “get rid of” the coefficient.

Ex.4x + 8 = 20Ex.9x – 7 = 2

Ex.7 – 3x = 4Ex.–13x – 1 = –1

Ex.4x + 6 = 9Ex.

11.4GENERAL EQUATIONS: PART II

Objective A - Solving an Equation of the Form ax + b = cx + d

Ex.5x + 7 = 2x + 16

What’s different about this equation compared to the others we’ve solved?
There are variable terms on BOTH sides of the equal’s sign.
Need to “move” one of them to the other side of the equal’s sign.

How do you “move” a variable from one side of the equal’s sign to the other?
How do you know WHICH one to move?

ANSWERS:
Move the one that has the SMALLER coefficient, you will avoid dealing with a negative number afterwards.

With this in mind, let’s look at 5x + 7 = 2x + 16again.

Ex.5x + 7 = 2x + 16which is the smaller coefficient, 5 or 2? Move that term.

Ex.3x + 6 = –23 – 2xEx.9 – 4x = 11 – 5x

Ex.9 + x = 2 + 3xEx.2x + 6 = 7x + 6

Objective B - Solving an Equation Containing Parentheses

You may have parentheses on one or both sides of the equal’s sign.
If that happens, you must use the Distributive Propertyto remove the parentheses.

Then you can solve the equation as you did in Objective A.

Ex.3x + 2(x + 4) = 13Ex.3x – 4(x + 3) = 9

Ex.4 – 3(x + 2) = 2(x – 4)Ex.2x – 7(x – 2) = 3(x – 4)

11.5 TRANSLATE VERBAL EXPRESSIONS INTO MATHEMATICAL EXPRESSIONS

Objective A - Translating When You Are Given the Variable

If you were taking a French course, you would need to be able to translatewords from English into French and from French into English.

The reason it’s helpful to study math is that, in real life, it enables you to find numerical values that you may need; these values might relate to mileage, salary, costs, etc...

In these real life situations, you need to be able to convertthe words into mathematical statements.

In this section, you will learn how to do some of these simple translations. You need to know what words mean to add, subtract, multiply, or divide.

Please refer to your textbook for a more complete list of translations. Below are some of the more confusing ones:

Operation / English / Expression
Addition / the total of 6 and x / 6 + x
y increased by2 / y + 2
Subtraction / z minus 3 / z – 3
6 less than x / x – 6
y decreased by1 / y – 1
difference betweenx and 5 / x – 5
subtract4 fromx / x – 4
Multiplication / twicex / 2x
one-third of y /
Division / x divided by 3 / or
the quotient ofa and b / or
the ratio of n to 7 / or

Translate the following into a mathematical expression:(Just translate word-by-word.)

Ex.z divided by tenEx.seven plus nine

Ex.the product of –3 and yEx.the sum of x and the product of 5 and y

Ex.the total ofaand the quotient of aand 7

Ex.the quotient of 3 less than z and zEx.the product of y and the sum of y and 4

Note:If you seem to have two operations next to each other with nothing in between (as in the previous example) you need to include parentheses.

Ex.The difference between the square of rand r

Objective B - Translating When You Are NOT Given the Variable

If you are asked to translate, “a number increased by 12” and you try to do so word-by word as we did in Objective A, you immediately run into a problem?

In the previous examples the variable was given to you; in this section, you need to assign a variable to quantity you don’t know.

For this example, you don’t know what “the number” is, so you FIRST write:

Let x = the number

Now, when you run into the words, “the number,” you can translate it to the variable x.

(You may use any variable.)

Now you can do the translation.

Ex.five less than some numberEx.the quotient of thirteen and a number

Ex.the ratio of a number to nineEx.six less than the total of three and a number

Ex.the quotient of six and the sum of nine and a number

Ex.the difference between 10 and the quotient of a number and two

11.6 TRANSLATING SENTENCES INTO EQUATIONS AND SOLVING

Objective A - Translating a Sentence into and Equation and Solving It

Once again, you will need to translate English into Math.
The following words get translated into an equals sign (=):

equals

is equal to These ALL mean =

amounts to

represents

Process:
1. Once again, use a “LET” statement.
2. You will be writing an equation.
3. Then you will solve the equation.
4. State the answer in a sentence.

Ex.The sum of seven and a number is three. Find the number.

Let =
Equation:
Solve the equation:
State the answer in a sentence:

Ex.Three-fifths of a number is equal to twenty. Find the number.

Let =
Equation:
Solve the equation:
State the answer in a sentence:

Ex.Three times the sum of a number and six is twelve. Find the number.

Let =
Equation:
Solve the equation:
State the answer in a sentence:

You will need to know these steps without being prompted with the clues I’ve typed above.

Ex.The total of a number divided by four and nine is two.

Objective B - Application Problems

In these problems, you need to use a “LET” statement to “let x = what you don’t know.”
Then you will write the equation, solve it, and state the answer.

Please note: These problems are simple enough so you may be able to find the answer without using the algebraic process.

HOWEVER, THE PURPOSE OF THIS COURSE IS TO TEACH THAT ALGEBRAIC PROCESS, SO THAT IS THE SOLUTION I WANT TO SEE BOTH IN THE HOMEWORK AND ON THE TEST.

Ex.As an electrical engineer, Anh Nyuang is a paid a salary of $962 a week. This is $43 more than she was paid last year. Find weekly salary that Anh was paid last year.

Let =
Equation:
Solve the equation:
State the answer in a sentence: