Linear Equations with Two Variables - Part 1

Objectives

After completing this lesson, you will be able to:

• Recognize a linear relationship.
• Given a formula, a table of values, a graph or a situation that represents a linear equation, identify the intercept and the slope.
• Given a table of values, a graph or a situation that represents a linear equation, find the formula.
• Given a formula, a table of values or a situation that represents a linear equation, draw a graph of the linear relationship.

Introduction

x = Hours Worked 0 1 2 3 4 y = Dollars Earned 0 10 20 30 40

If Julie earns $10.00 per hour, the table and graph above show a relationship between x, the hours she worked, and y, the dollars she earned. When the value of x increase by 1, the value of y always increases by 10. So the rate of change is 10 and is a constant.

In the lesson on Direct Variation, you saw several examples where the change reason is constant. Since all the direct variation examples satisfy y = kx, where k is constant, they all include the point (x, y) = (0,0).

The following linear relationship example is similar, but point (x, y) = (0,0) is not included, therefore it is not a direct variation relationship.

x = Weeks Elapsed 0 1 2 3 4 y = Dollars in Bank 20 30 40 50 60

If Mark opens a bank account with $20.00 and deposits $10.00 every week, the graph above shows the relationship between x, the number of elapsed weeks, and y, the dollars Mark has in the bank.  Some observations:

• Mark begins with $20 dollars (i.e., when x = 0, y = 20).  Another way of saying this is the y-intercept is 20.
• When x increases by 1, y increases by 10.  Another way of saying this is the rate of change is equal to 10 or that the slope is 10.

Linear relationships are similar to direct variation relationships since the rate of change (slope) is constant in both, but unlike direct variation relationships, linear relationships can include point (0,k) where k is different from 0.

Definitions

Given a relationship between two variables x and y, the relationship is linear if the rate of change is constant, i.e., when x increases by 1, y always increases by a constant value.

• The constant in which y increases when x increases by 1 is called the slope and is generally expressed by the letter m.
• The value of y when x = 0 is called the y-intercept and is generally expressed by the letter b.

If you begin with 15 cookies and one cookie costs 20 cents, the table above shows the relationship between x, the amount you spent, and y, the number of cookies you have. Some observations:

• You begin with 15 cookies (i.e., when x = 0, y = 15). Another way to say this is the y-intercept is 15.
• When x increases by 1, y increases by 5. Another way to say this is the slope is 5.

Rate of Change and Slope

• When we say that x increases by 1, we can also say: run = $\mathrm{\Delta x}=1$.
  
• When we say that y increases by m when x increases by 1, we can also say: rise = $\mathrm{\Delta y}=\mathrm{m}$.

In the above application, complete the following steps:

1. Move the point on the far right of segment Δx to the left until the run Δx is equal to 1. You should see that rise Δy is equal to 2. So that every increase of 1 in x, y increases by 2.
2. Now move the point until the run Δx is equal to 2. In order for the slope of the line to remain the same, we must raise the point on y up until the rise Δy is equal to 4.
3. Note that by saying "for each increase of 1 in x, y increases by 2" is the same as saying, "for each increase of 2 in x, y increases by 4".
4. If m = 0.5, find three different combinations of Δx and Δy which are consistent with this slope.
5. If m = -2, find three different combinations of Δx and Δy which are consistent with this slope.

The following table show the relationship between run and rise for this line:

Use the above application to find runs and rises consistent with slopes of 3, 1, 0.5, -0.5, -1 and -2.
At this time, you should be able to understand the following relationships associated with slopes

• $m=\frac{\mathrm{rise}}{\mathrm{run}}=\frac{\mathrm{\Delta y}}{\mathrm{\Delta x}}$
• rise = m × run or Δy = m × Δx

When the slope is a fraction, it is not always convenient to think of the run as being equal to 1. In these cases, it helps to remember that $m=\frac{\mathrm{rise}}{\mathrm{run}}=\frac{\mathrm{\Delta y}}{\mathrm{\Delta x}}$. For example, consider that $m=\frac{2}{3}$

So $m=\frac{2}{3}$, with each advance of 1 in direction x, we raise $\frac{2}{3}$ in direction y. Also, with each advance of 3 in direction x, we raise 2 in direction y. So given a fraction slope: $m=\frac{\mathrm{\Delta y}}{\mathrm{\Delta x}}$. Then, if the run is Δx, the rise is Δy.

Use the following application to practice how to respond to questions related to slopes:

Finding the Formula for a Linear Relationship

Examples:

The following interactive application will allow you to find formulas for linear relationships. Use the sliders to change the values of b, the y-intercept, and m, the slope. Then click the button to build the relationship and reveal the formula.

Generalization: Consider a line that has a y-intercept of b and whose slope is m. A y-intercept of b signifies that when x = 0, y = b. A slope m means that every time x increases by 1, y increases by m units. Beginning at point (0,b) and increasing y by m units each time x increases by 1 produces the following table:

This same table can be rewritten expressing y as b plus the number of m's that have been added.

It should be made clear that the values of x and y have not changed in these tables. However, on the second table, we can quickly see that y = b + something × m. This observation permits us to conclude that the something is in reality x . So the relationship between x and y can be expressed as y = b + m × x. Another representation of this is
y = mx + b,

which is one of the general formulas for a linear relationship.

Graphing Linear Relationships

Examples:

Example 1: Plot the graph of the linear equation y = 3x + 2. Since the y-intercept is 2, we know the line passes through point (0,2) and since the slope is 3, the line increases 3 units for each unit the line moves to the right, as shown in the following figure:

Example 2: Plot the graph of the line which has a y-intercept of 1 and a slope of $\frac{5}{2}$. Since the y-intercept is 1, we know the line passes through point (0,1) and since the slope is $\frac{5}{2}$, the line raises $\frac{5}{2}$ units for each unit the line moves to the right. Then, for every two units the line moves to the right, the line raises $2\times \frac{5}{2}=5$ units. This is equivalent to saying that for each two units the line moves to the right, the line raises 5 units, as shown in the following figure:

Example 3: In the application shown below, complete the following steps:

1. Move the slider labeled Slope so the slope is equal to 2
2. Move the slider labeled y-Intercept so that the intercept with the y axis is equal to 3

   Click on the button to create a table with evaluations of the function

3. Verify the graph which results has a y-intercept of (0,3) and for every increase of 1 in the value of x, the value of y increases by 2
4. To continue practicing, use the application to express the following lines and verify for each that the slope and y-intercept are correct:
   1. y = -2x + 4
   2. y = x - 3
   3. y = -5x + 2
   4. y = -2x - 3

Practice: In the application shown below, complete the following steps:

1. Move the point to the change y-intercept on the line.
2. Move the second point to a point consistent with the slope.
3. If the graph is correct, the message, You are correct! should appear.
4. For a new formula, press the New Equation button.

Summary

Now that you have completed this lesson, you should be able to:

• Recognize a linear relationship.
• Given a formula, a table of values, a graph or a situation that represents a linear equation, identify the y-intercept and the slope.
• Given a table of values, a graph or a situation that represents a linear equation, find the formula.
• Given a formula, a table of values or a situation that represents a linear equation, draw a graph of the linear relationship.