Marked June 7th

Linear Relations page by: Jas, Maggie, Jesse,

What is Linear Relations?

Linear Relations is the relationship between two variables that appears as a straight line when graphed. Linear relations is broken down into linear and non-linear, and direct and partial variation. These can be identified using equations, graphing, and first differences.

Definitions and Equations

- The steepness determined by vertical change and horizontal change
Y-intercept - where the line crosses the y axis
X-intercept - where the line crosses the x axis Linear Relation
Analytic Geometry - the study of lines
Linear - a straight line
Non-linear - not a straight line
Standard Form- An equation written in Ax+By+C=0 format

Basic Rules:

When using standard form there are three rules that must be followed at all times.
They are:

  1. All terms must be on the left side
  2. The first term must be positive
  3. There can't be any fractions or decimals in the formula

An example of a proper equation that has been converted into the standard form would be:
From y=mx+b form to Standard Form:

y =3x+12 ...........ymx+b form to
3x+y-12=0.............Standard form

Linear or Non-Linear


  • Examine the equation, if there are any exponents 2 and over it is not linear!
  • You can check by looking at the graph, if the line is straight then it is linear, if it is not straight then it is not linear
  • You can also check by looking at the first differences. If the First differences are the same or follow a pattern then it is linear, if the first differences are all different or follow no pattern then it is non - linear.


Using First Differences:

  • You must first make sure that your "x" column in your table of values increases by the same amount each time. If your table does not do that then you have made a mistake
  • To find first differences you must subtract the "x" column from the "y" column. This will give you your difference between the two columns.
  • If the first differences are all the same or have the same pattern then you know that your line is linear. If your first differences are different or have an irregular pattern then you know that the line is non-linear.




First Differences
3 – 1 = 2
5 – 2 = 3
7 – 3 = 4
9 – 4 = 5
11 – 5 = 6

This is an example of a linear line. We can tell that this is a linear line because the first differences in the chart are in a constant pattern of increasing by one.

Height of Building (m)
Number of Stairs
First Differences
1000 – 800 = 200
700 – 600 = 100
400 – 250 = 150
100 – 90 = 10
60 – 30 = 30


Formula's And When To Use Them

There are three major formula's that we will be focusing on in grade 9. The first one is Y=MX+B. This one is the most commonly used out of the three and is very useful when graphing. In this formula, M repersent the slope, and B repersents the y-intercept. When asked in a question to find the equation of a line, the first thing you would write down is Y=MX+B.

The next formula is the standard form formula,Ax+By+C=0, which isn't used much in grade 9 math but more in grade 10. The only thing we would use it for in grade 9 is if you where asked to convert a Y=MX+B formula into standard form or if you wanted to find the Y and X intercepts of a line.

The last formula is Y2-Y1/X2-X1 which would be use if you were given 2 coordinates and asked to find the slope. If you are given 2 coordinates and asked to find the equation of a line you first write y=mx+b. Then you need to find M and as you already know, M is the slope. To find M when given 2 coordinates use the formula M=Y2-Y1/X2-X1. Once you have solved that you would want to find b. to find b use the formula y=mx+b and use one of the coordinates. Sub in the y coordinate to y and the x coordinate to x, than solve you should get b. Then sub M and B into y=mx+b and that is how to find the equation of line.

The last formula is RISE/RUN. This formula is used when you are given a line on a graph and you are trying to find the slope. Simply find the measurments it takes to get from on point to the other, going across and up/down. For example,


We will be doing some examples for each formula starting with M=Y2-Y1/X2-X1.
Example 1. Find the slope of a line that passes through points (5,3) and (4,9).
So the first thing you would write down is the formula. M=Y2-Y1/X2-X1.
Then you would label one coordinate A and the other B. A(5,3) and B(4,9)
After that get the Y coordinate from coordinates B, sub it into Y2. Get the Y coordinate from coordinates A, sub it into -Y1. M=4-5/X2-X1
Now do the same thing with the two X coordinates. M=4-5/9-3
Last solve. M=-1/6 is the slope of the line
In this example we will be using the formula Y=MX+B.

Example 2. Write the equation of a line that goes through the point (0,-3) and has a slope of 5.
For this all you need to do is fill in M and B, M is the slope so M is 5. Y=5X+B
Then fill in B which is the y-intercept (the y coordinate). Y=5X-3 is the equation of this line.

For this example there isn't much to show but we will be using the formula Ax+By+C=0.
Example 3. Convert this into standard form Y=7X-9.
First you need to move every thing to the left, and make sure you change the sign when you move it. -7X+Y+9=0
Then if the first term on the left is negative you need to multiply the whole equation by negative one. -1(-7X+Y+9)=-1(0)
Once you have multiplied the formula by negative 1 the final answer will look like 7X-Y-9=0
For this example we will be using RISE/RUN.

Example 4. Find the slope of the black line.

So after you have got your line, you will find the 2 points, put one at the bottom of your line and one at the top.Count the number of points it takes to get from the bottom point to the top point going upwards therefore this is the RISE. Once you have found your rise you must find your run, to do this you will have to start at the bottom points and count how many points it takes to get to your other line going across therefore this is called RUN. Put them in your formula and do the math, lets say that your rise was 6 and your run was 2 using the formula RISE/RUN your answer would be 3.

There is one more example for the formula Ay+By+C=0 and this is how to find the Y and X intercepts of a line.

example: 4x+8y+16=0

In Real Life

Unlike some other units in math, Linear relations is actually very common in everday life. Infact, most people use it without even knowing. In application to the real world, linear relations usually reffers to the relation between time and something. That somthing could be distance,earnings,pionts totaled, etc. It basically shows that progress of a variable over a period of time. For example:

Kevin Garnett earned 20 million dollars over one season in the NBA. He lost 5 million dollars to the NBA after he was fined for misconduct to a fan. He earns a bonus 15 million in endorsments over his whole careers regardless. The Boston Celtics promise him 20 million dollar each year he decideds to play for them. How much will Garnett earn in his remaining years before he is retired?

This is a perfect example of a linear relation, so lets put it into Y=MX+B format. Wheather Garnett decides to play or not, he will earn atleast 15 million dollars more in endorsments buy the end of his career, but because of the fine he losses 5 million, so he is garunteed 10 million(10,000,000). This is were Garnetts salary will start at no matter what, and therefore it is the Y intercept, and our B (in Y=MX+B). From that point on, Garnetts salary can increase on a 20 million dollar for each year he plays, so the slope is 20 million (20,000,000). X repersents how many more years he plays. When the numbers are plugged in you get Y=20,000,000X+10,000,000. If Garnett played 4 more years, this is how you would find his earnings:


Therefore Garnett would have earned $90,000,000 more dollars in 4 years before he retired.

Linear relation graphed.

Finding the y-intercept in a Y=MX+B formula is simple, as the B is the y-intercept (remebering that if there is no B then the y-intercept is therefore 0).
Although finding the y-intercept when the Linear eqaution is written in standard form is not as easy. Lets use the following equation to help explain the procedure, 4x+8y+16=0. To find the y-intercept, you have to solve the equation for y (meaning you must isolate y and find out what it equals). The first thing you do to find y is make X=0. X must always equal zero when trying to find y, because if you solve for one variable while there is still another variable in the equation you will get a number attached to the other unsolved variable, which dosen't give you an exact answer. Therefore X=0. Lets see how this is done on our equation
4(0)+8y+16=0.<--- notice how the X has been repalced with a 0.
8y+16=0.<---after multilpying 4 and 0 you get 0, which dosen't need to be written.

The next step is to isolate y. To isolate y we must first get rid of the C (16).
8y+16-16=0-16.<---to get rid of the 16, we subtract 16 ffrom that side and the other side, because whatever you do to one side of a equation you must do to the other.
8y=-16.<---after subtacting 16 from both sides we are left with this equation.

The last step is to split the 8 and the y. To do this we have to subtract both sides by 8.
8y/8=-16/8.<---dividing both sides by 8 to isolate 8.
y=-2.<---after divding bith sides by by 8 we are left with this.

Now that y is completly isolated, and solved, you have the y-intercept, which equals -2.


Follow the same steps for finding the Y-intercept, except now Y=0 and you isolate for X.


The last aspect to Linear relations is Graphing. With graphs, it goes either one of two ways; your either making the graph, or you use the graph to solve questions regerding it. The reason the graph is important is because it helps to visually see the linear relations, and provides other options, such as a line of best fit, which help you further understand the linear relation.

Graphs help us do so many things in various ways, but we're just going to focus on how to graph linear realtions, and understand them.

When it comes to graphing Linear relations,what you are going to graph is the corridinites or the LOBF (Line Of Best Fit). Graphing this line is as simple as just connecting the dots in a stright line or makeing a line that is as close as possible to all sets of data. Although, the tough part is to find the co-ordinates and get them graphed. To find the coordinates you have to fill in the blanks to the Y=MX+B formula, giving you the slope and y-intercept. Once you have that you can graph the y-intercept and then using the slope graph the co-ordinate giving you the line on the graph.

The difficult parts of graphing are using equations, formula's, and finding the linear relation. The rest of graphing is the basics things to remeber when making the graph. They are:

  • Rember your titles. One title for the graph, and one for the x intercpt and y intercept if required. Also label the axis (x and y).
  • Label the lines on the graph with there equations (ex.Y=5x+3)
  • If there is a graph were you have to label the axis, always remeber that time is on the x-axis.
  • Make sure all lines are straight (if not they wouldn't be linear therefore there not linear relations).
  • Draw arrows at the end of lines to indicate that the line keeps going.

Here are you examples of linear relations being graphed. The first is on a cartasian plain, the second is showing the relation ship between time and distance on a standard graph.

Equation for first Linear relation graph-Y=2X+4.


This equation and graph is showing the Linear relation between time and distancewith the equation Y=10X


The slope of a horizontal line is zero
The slope of a verticle line is undefined
If a line is steeper it will have a larger slope

Citations: Hendricks, J.D. (2008) MPM 1D1 - Course Notes