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Many mathematical models are represented in many different ways. For example, using a table, graph, picture, words or using an algebraic equation written in a general algebraic form.  In the previous activity, you were introduced to one algebraic form of a linear relation, y = mx + b, where x and y represent the coordinates of a point on the line with a slope equal to the value of 'm' and the y-intercept equal to the value of 'b'.    

Working with algebraic equations often involves substituting one or more values into the equation to solve for a value that is not given.

This is the Portfolio icon. Word Journal

Open your Word Journal.

In the first column, write the word “solve”.

In the second column write down some everyday meanings for the word.

Leave the third column empty for now.

Save to your Portfolio.

 

The equals sign in an equation tells you that the value on the left side of the equal sign is the same as the value on the right side of the equal sign, even when they have a different appearance.

2 + 3 = 4 + 1

This is true whether the equation uses numbers, or variables, or a combination of numbers and variables. The equal sign tells you that the two sides of the equation are in balance. As you solve for a variable, it is very important that you maintain the balance in the equation by working with both sides at the same time.

In the first video, a mathematical equation is written to represent the objects in the balance. The solution demonstrates the use of opposite operations to solve for the variable:

 

Watch the following video to view the use of the four operations, addition, subtraction, multiplication and division to Solve One Step Equations.

You probably were able to determine the answer to many of the examples simply by inspection. As the equations become more complex, you may find that more steps are required and the answer is not available by inspection.

Linear equations may require the use of 2 steps to solve for the variable. In the following video, the solution of a 2 step equation is modelled using a balance beam.

 

As the number of terms (definition:a single number or variable, or numbers and variables multiplied together, separated by + or - signs) in an equation increases, the number of steps to isolate the variable also increases. You will continue to use the four operations to isolate the variable.

Watch the following video that demonstrates the solution using a balance and shows the solution of a linear equation that starts with 4 terms.

 

In the examples that you have seen, most of the values in the equations are postitive integers (any one of the numbers …, -3, -2, -1 , 0 , 1, 2, 3…). The coefficients(definition:the factor by which a variable is multiplied) and the constants can be positive or negative integers, fractions, or decimal values. 

In the following video, a range of values will be used for the coefficients and the constants. Notice that the general process of isolating the variable and solving remains the same.

 

Action.

ACTION

This is the Portfolio icon. Mathematical Processes

In this activity, the Mathematical Process Representing is the focus.

Open your document U2 Mathematical Processes and read the descriptions for the process.

As you complete the activity, notice when you are accessing the process and insert your record below the description of the process.

 

In Activity 5, you were graphing a linear relation using different pieces of information about the line.  When an investigation is done and data points are collected, the pieces of information that allow you to create an equation of the line are based on the points or the characteristics of the line of best fit.

Writing the Equation of a Line

In the previous activity, you started with the equation and learned that the values in the position of m and b in the equation, y = mx + b, are the slope and y-intercept. You used the information to graph the line.

In this activity, you will be writing the equation of the line using the slope and y-intercept values. You will be exploring the different ways that the key information can be provided about the line. In each case, you will determine the values for the slope and the y-intercept and then use them to successfully write the equation of the line.

Check your understanding of identifying the values by competing this activity from Khan Academy.

Writing the Equation given its Graph

slopeQuests

Long Description

 

Writing the Equation Given its Slope and y-intercept

Using the slope and y-intercept, decide which equation represents the linear relation.

1. The slope is 5 and the y-intercept is 2/3. Which equation represents the linear relation?
a. y = 2/3x + 5
b. y = -5x + 2/3
c. y = -2/3x + 5
d. y = 5x + 2/3
e. y = -2/3 x - 5

Answer

d: y = 5x + 2/3

 

2. The y-intercept is 3 and the slope is -4. Which equation represents the linear relation?
a. y = -4x - 3
b. y = -3x - 4
c. y = -4x + 3
d. y = 4x + 3
e. y = 3x - 4

Answer

c: y = -4x + 3

 

3. m = 1/2 and b = 4. Which equation represents the linear relation?
a. y = 1/2 x + 4
b. y = -4x + 1/2
c. y = 4 - 1/2 x
d. y = -4x - 1/2
e. y = -1/2 x + 4

Answer

a: y = 1/2 x + 4

 

4. b = -2 and m = -3/4. Which equation represents the linear relation?
a. y = 3/4 x + 2
b. y = 3/4 x - 2
c. y = -3/4 x - 2
d. y = 2 - 3/4x
e. y = -2x - 3/4

Answer

c: y = -3/4 x - 2

 

5. The slope is -4 and the y-intercept is 7. Which equation represents the linear relation?
a. y = 7 x - 4
b. y = 4x + 7
c. y = 4x - 7
d. y = -4x + 7
e. x = -4y + 7

Answer

d. y = -4x + 7

 

6. The y-intercept is -5 and the slope is 3/5. Which equation represents the linear relation?
a. y = 3/5 x + 5
b. y = -3/5 x -5
c. y = 3/5 x - 5
d. y = -5x - 3/5
e. y = 5x + 3/5

Answer

c: y = 3/5 x - 5

Writing the Equation of a Line given its Slope and a Point on the Line

If you know the slope of the line and the location (coordinates) of any other point on the line, you still have enough information to graph the line or to generate its equation. You do not always need to know the y-intercept. You can see what this could look like using a table of values or a graph.

Find the equation of the line with slope, m = 2 that passes through the point (3, 7).

You can begin by creating a table of values with a difference column and including the given point, (3, 7).

A table of values with x and y values. The x values include -1, -, 1, 2, 3, and 4 while only the y value 7 is matched to the x value 3.


Think about what you know about the slope and y-intercept:

  • What does the sign of the slope tell you?
  • What is the value of the x at the y-intercept?
  • Imagine the of a line that has a slope of 2 that passes through the point (3, 7).  Where do you think the y-intercept will be?
  • What does that tell you about the values you need to add to your table of values?
An image of a grid with the point (3,7) plotted.

chart1

Long Description

 

You can use this information to move in both directions from the point and fill in the table of values.

From the mental image that you have of the graph, you know that the y-intercept will be to the left and below the given point since the slope is a positive value.

An image of a grid with the point (3,7) plotted.

In other words, you will need to move to the left and down to get to the y-intercept.

You can see that to move from (3, 7) back to (0, 1), you used the slope ratio three times - starting at (3, 7), where x = 3 to move back to where x = 0, run = -3, you have 3 horizontal changes of 1, so you need 3 vertical changes of 2, rise = 3 x -2 = -6.

An image of a table of values. The values for x are -1, 0, 1, 2, 3 and 4. The values for y are -1, 1, 3, 5, 7 and 9.


The interactive below will help you see how you can use the slope to move down (-2) and to the left (-1) 3 times to arrive at the y-axis where x = 0.

slopeAnim

Long Description

 

Using this information you can now write the equation of the line. 

y = mx + b

m = 2 and b = 1

Substituting the values, y = 2x + 1

In a similar manner, you can use the general equation,  y = mx + b, to determine the unknown value of the y-intercept, b, using an equation-solving process. In these cases, you will know three of the values, x, y, and m, that you will substitute into the equation and then solve for the unknown value, b.

 Follow the process of determining the value for the y-intercept and writing the equation of the line when you have the slope and a point on the line:

 

solveForB

Long Description

 

For further study, view the presentation Finding the equation when you have point and slope.

Writing the Equation of a Line given Two Points on the Line

Situations occur where you will only have 2 points that are on a line and you will need to determine both the slope and the y-intercept before you can use the write the equation in slope y-intercept form,  y = mx + b. You can look at the slope calculation in a table of values.

From the table, you can determine the change in the y values and the change in the x values.

A table of values with x values of -2 and 2 and y values of -5 and 1.


The change in y is the difference between 1  and -5 , or 1 - (-5) which equals 6, and the change in x is the difference between 2 and -2, or  in x is 2 - (-2) which equals 4.

A table of values with x values of -2 and 2 and y values of -5 and 1. The difference of 4 is shown between x values and the difference of 6 is shown between y values.


With the values for the change in y and the change in x, you can calculate slope.

An image of a slope equation with rise being 6 and run being 4 and the overall slope being 3/2.


Recall that earlier in this unit, you saw a number of definitions for slope and you probably notice that some are being used more often than others.

Currently you are seeing the use of:

An image of the slope equation indicating that slope equals rise over run which is equal to change in y over change in x.

 

Follow the process of determining the slope and y-intercept and writing the equation of the line when you have two points on the line:

 

You do not need to use the notation of y2 - y1 or x2 - x1 in your work. This notation is another way of saying rise and run.

eqnWithPoints

Long Description

 

When you have integer coordinates, graphing to determine the slope and/or y-intercept is a good strategy. However when you have fractional or decimal coordinates, graphing becomes more difficult and unreliable. Using algebraic methods to determine slope and y-intercept values is quicker and more accurate.

For further study, view the presentation Finding the equation when you have 2 points on the line from Homework Help.

For a review of the slope y-intercept form of a linear equation, try out this following activity from Khan Academy.

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Complete U2A6 Assignment.

 
Consolidation

CONSOLIDATION

You have explored the algebraic representation of a linear relation and can connect the slope and the y-intercept with the values in the equation, y = mx + b. You have used different information about a line to draw the graph and to write the equation.

 

This is the Portfolio icon. Math Processes

In this activity, the Mathematical Processes Representing was the focus.

Open your document U2 Mathematical Processes and complete your record of when you used the process of representing.

Save to your Portfolio.

 
 

Congratulations, you have completed Unit 2, Activity 6. You may move on to Unit 2, Activity 7.

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