Algebra

Literal numbers can be used to represent either variables (the value of the letter can change) or constants (the value of the letter never changes). We have all used algebra at some point in our lives and were probably exposed to it at an early age.

Example

For example, we all learned the area of a rectangle is equal to the width (w) times the height (h), while in algebraic terms this equation looks like the following:

A= w x h

So once we know the width and the height we can solve the equation.

A term is an algebraic expression involving letter and/or numbers (called factors) which are multiplied together.

Example

3x
Is an algebraic expression, with a single term which has the factors 3 and x in it.

Example

3x + 6y
Is an algebraic expression which has two terms in it 2x and 6y.

Example

8x and 6x are like terms

4x² and 22x² are also like terms

7y² and 3x² are not like terms because they do not have similar variables.

Remember we can only add or subtract like terms. Now lets try an example:

Example

Simplify the following:

8x + 6y – 3x + 9a
= 5x + 6y + 9a

Notice that the only like terms in this problem are 8x and -3x. we cannot do anything with the 6y or 9a terms. So we group together the terms we can subtract, and just leave the rest.

Example

Let’s try another one:

Simplify -6{-3(2x-3y) + y]

Hint: go back to the order of operations if you are not sure what to do first.

-6[-3(2x-3y) + y]
= -6 [-6x + 9y + y]
= 36x – 54y +y
= 36x – 53y

Example

Multiply the following: x²(x³ + 5y)

Let’s start by multiplying everything inside the brackets by x2

= x6 + 5y x²
We can not simplify this equation any further, so we present our answer as it is.

Example

Let’s try another one:

Multiply the following: (4x +3)(y + 6)

First we take each term of the first bracket and multiply them by the second bracket:

(4x +3)(y + 6)
= 4x(y + 6) + 3(y + 6)

Next we expand each bracket:

(4x +3)(y + 6)
= 4x(y + 6) + 3(y + 6)
= 4xy + 24x + 3y + 18

We cannot simplify this answer any more since there are no like terms so we are done at this point.

Example

Let’s try one more:

Multiply (x – 6)²

Let’s start by expanding this problem as such:

(x – 6)²
= (x+6)(x+6)

Now let’s multiply the first bracket by the second bracket just like we did in our last example..

(x – 6)²
= (x+6)(x+6)
= x(x+6) + 6(x+6)
= x2 + 6x + 6x + 36

Notice we now have two like terms 6x and 6x so we can simplify this problem even more.

(x – 6)²
= (x+6)(x+6)
= x(x+6) + 6(x+6)
= x2 + 6x + 6x + 36
= x² + 12x + 36

Example

Simplify the following: (5xy)(8x²y³) / 10x³y²

The first step is to simplify the top part of the equation. Once this is done we get the following:

(5xy)(8x²y³) / 10x³y²
= 40x³y⁴ / 10x³y²

Now you can reduce like terms from the top and bottom. For example 40x³ divided by 10x³ will give you 4 and y⁴ divided by y² will give you t² so your final answer should look like this:

(5xy)(8x²y³) / 10x³y²
= 40x³y⁴ / 10x³y²
= 4y²

Example

Let’s try another example:

2x²y – 6xy / 2x

In this example we will divide the problem into two fractions, both with te denominator 2x

(2x²y/2x) – (6xy / 2x)

Now we will reduce each side such that we get the following:

2x²y – 6xy / 2x
= (2x²y/2x) – (6xy / 2x)
= xy – (6xy / 2x)
= xy – 3y

Example

Let’s do an example:

(4/5) / (6/y) = (4/5) x (y/7) = (4y/35)

Example

Lets try another example, I will show you two different ways you can solve this problem and you can decide for yourself which one is easier.

Simplify the following: (4 + 1/x) / (4/x + 4)

Solution 1: multiplying by the reciprocal:

Solution 2: multiplying the top and bottom

This process is aiming to get the letter on the left hand side of the equation by itself. To do this we must balance whatever we do to one side of the equation with the other. So for example if you add 4 to one side of an equation you must also add four to the other.

Example

Let’s look at some examples:

Solve this equation: x + 7 = 12

Remember we want to isolate the x on one side of the equation. So on the left hand side we get x and on the right we get 12 minus 7. You final answer will be x equals 5. Here are what the steps look like:

x + 7 – 7 = 12 - 7
x = 5

To check your work all you have to do is substitute your answer back into the original equation for x.

5 + 7 = 12

Example

Let’s try this one:

Solve the following: 12x = 96

Lets start by thinking about what we are trying to answer. In this case we want to know what number times 12 will equal 96. You might be able to figure this out in your head, but let’s go through the steps anyway.

12x = 96

X = 8

Now let’s check our work. 12 x 8 = 96 is correct so we have done this problem correctly.

Example

Let’ try one more!

Example

Let’s look at a few examples.

Remember that the area of a rectangle is equal to the width times the height.

A = w x h

Express the following formula in terms of width.

To solve this equation we need to get the width term (w) on one side by its self. So we can divide both sides by the height term (h) to get eh following solution:

A = w x h
 

Which will reduce to:

So why might you want to do this? Well lets say you knew the area and heights for many rectangles and wanted to get the width of each, well by manipulating the formula we can now quickly solve the equation for the variable we are most interested in.

Example

Let’s try another example:

Solve the following for Y1: Y = 6(Y₂ – Y₁)

In this example we will start by dividing both sides by 6.

Next we add Y1 to both sides and then subtract Y/6 from both sides:

Now we have solved this equation for Y₁, and we see that it is equal to Y₂ minus Y divided by 6.

Here is a general outline of how you can tackle word problems:

1. Read the problem carefully, try to get a feel for the whole problem, and see what information you have and what information you need. You should write down the questions you are being asked to solve. Figure out what information you were given in the problem and what information you need to know in order to answer the question.
2. Next work in an organized manor. Label all your variables, draw and label any pictures or graphs and explain your reasoning as you go along.
3. The third step is to look for key words. Certain words will indicate certain mathematical operations. For example increased by, more than and added to all imply you will need to do some sort of addition, where as out of, percent or ratio imply you will need to do division.
4. Next you will form an equation and solve the equation. In this case you could form an equation based on the problem or apply an equation you already know.
5. Lastly you should check your answer and write a sentence with your answer (remember to include your units).

Solve the Following:

Two streams have a combined flow rate of 32.4 cubic feet per second. We know that one stream has a flow rate that is 5 cubic feet per second higher than the other. What is the flow rate of the two streams?

Solution 1:
If we assume x is equal to the flow rate of the slower stream than the flow rate of the faster stream will be equal to

(x + 5) ft³/sec

We also know that together the two streams will release the following:

x + (x +5) = 32

So we get:
2x + 5 = 32

2x = 27
x = 13.5

When we solve for x we get a flow rate of 13.5 ft³/sec and since we know that the second stream has a flow rate which is 5 ft³/sec faster we can add 5 to the first flow rate of 13.5 to get 18.5 ft³/sec.

Now let’s check our work. If we add together the tow flow rate we should get a combined flow of 32 ft³/sec. Since 13.5 + 18.5 add up to 32 we know we have done this problem correctly.

Solution 2:
We could have also solved this equation by solving simultaneous equations. I will now show you how to do this using the same example.

Lets start by assuming that X is equal to the larger flow rate and Y is equal to the smaller flow rate.

We there for know that X + Y = 32 and X – Y = 5

Now let’s set up our equations like this:

X + Y = 32
X – Y = 5

If we simplify these two equations we see that we get:

2X = 37

And

X = 18.5