Equations and Graphs

We all need reminders from time to time about mathematical concepts. Take a look at the coordinate plane below, and refresh your memory about it.

You should recall that there are four quadrants labeled I, II, III, IV.
There are two different axes called x and y.
Points have coordinates, like point B on the graph, and are always indicated as (x, y).
The origin is where the x and y axes intersect, and has the coordinates of (0, 0).

coordinate plane labeled.png

If you are still feeling fuzzy on the coordinate plane, watch this video to further review the main ideas.

We'll come back to the topic of graphing and the coordinate plane again later.

For now, we want to look at finding a solution to an equation with two variables x and y. Our two variables can be listed as an ordered pair (x, y) and if they are a solution, the ordered pair will make the equation true.

The process is simple enough. You need to substitute the values of the variables into the equation and then check to see if the equation is true or false.

Example 1: Is (6, 3) a solution to the equation $8 x - 10 y = 28$ ?

Replace the x with 6 and the y with 3 and check.

8(6) - 10(3) = 28?

48 - 30 = 28?

18 = 28?

No, 18 $\neq$ 28. Therefore, (6,3) is NOT a solution to $8 x - 10 y = 28$ .

Practice problem 1:

Click the question prompt below to reveal the answer

Is (-3, 5) a solution to the equation $5 y + 4 x = 13$ ?

Yes, $5 (5) + 4 (- 3) = 13$

$25 - 12 = 13$

You can represent a relationship between two variables in several different ways. Let's take a look.

Example 2: James and David were born on December 11, but they were born 6 years apart. How can you represent the relationship between James' age and David's age in different ways?

You can create a table of values:

James' age	1	2	3	4	5	6	7	8
David's age	7	8	9	10	11	12	13	14

You can write an equation:

Let x = James' age

Let y = David's age

David (y) is always 6 greater than James (x). (See the table above to verify.) So $y = x + 6$ .

Finally, you can also draw a graph on the coordinate plane we discussed above. (Again, James' age will be x, David's age will be y.)

All three methods shown above represented the same numerical relationship in different ways. Did you find one or another to be more useful? You will find some to be more useful than others, depending on the circumstances.

Practice problem 2:

Click the question prompt below to reveal the answer

Jill and Molly are going to run at the track at their school. They were to meet at 4, but Molly was late. Jill had completed 4 laps when Molly finally arrived and immediately began running with Jill. If they run at the same pace and stop at the same time, how can you represent the relationship of the laps that they ran that day? Use a table, an equation, and a graph to represent the relationship.

Jill	4	5	6	7	8	9
Molly	0	1	2	3	4	5

x = Jill, y = Molly

$y = x - 4$

In the previous example and practice problem, we have seen tables and graphs that show noticeable patterns.

Inductive Reasoning is the process of reaching a conclusion based on an observed pattern. You can use inductive reasoning to predict values that aren't directly shown.

Example 3: Steve wants to put down new flooring in the hallway in his house. He wants to be very creative and use two different kinds of wood. For every piece of light oak wood he puts down, there are a total of 9 pieces of wood. What is the total number pieces of wood if Steve used 10 pieces of light oak wood?

Start off with a table of values:

Number of light oak pieces, x	Total number of pieces, y
1	9
2	18
3	27
4	36
5	45
6	54
7	63
8	72

Now, write an equation:

Since the table shows that you need to multiply the x-value by 9 to get the y-value, we will use the following equation to illustrate the table:

$y = 9 x$

Now, substitute the number of light oak pieces that Steve wants to use for the x-value.

$y = 9 (10)$

$y = 90$

Steve is going to use a total of 90 pieces of wood on the flooring in his hallway.

You could also use a graph to help solve the problem.

Practice problem 3:

Click the question prompt below to reveal the answer

Cassandra collects football and basketball player signatures. She noticed that for every 6 football player autographs she has, she has 1 basketball player autograph. What is the total number of basketball autographs Cassandra has if she has 48 football autographs. Use a table, graph and/or an equation to solve the problem.

8 basketball autographs.