PLTL #4 Solving Linear Equations                                Name ___________________________________

Overview: Describing Linear Relationships

 

We focus on four ways to mathematically describe linear relationships between two quantities.

 

1. Equations in Two Variables

 

Text Box: The general form of the equation of a line is Ax + By = C. Two very useful forms of the equation of a line are the slope﷓intercept form and the point﷓slope form.1. Write the equation of a line with a slope of ‑ 3 and a y‑intercept of (0, ‑4).   Answer:    y = ‑3x ‑ 4   

 

2. Write the equation of the line that passes through (5, 2) and (‑5, 0). Answer in slope‑intercept form.  Answer:

 

 

 

 

2. Rectangular Coordinate Graphs

 

Text Box: The graph of an equation is a "picture" of all of its solutions (x, y). Important information can be obtained from a graph.3. Complete the table of solutions for 2x ‑ 4y = 8. Then graph the equation.                  

 

 

 

 

 

 

 

 

 

 

4. See the graph below

 

a. What information does the y‑intercept of the graph give us? Answer: When new, the press cost $40,000

 

 

 

b. What is the slope of the line and what does it tell us?     Answer: ‑5,000; the value of the press decreased $5,000/yr.

 

 

 

 

 

 

 

3. Linear Functions

Text Box: We can use the notation f(x) = mx + b to describe linear functions.

5. The function f(x) = 25 + 35x gives the cost (in dollars) to rent a cement mixer for x days. Find f (3). What does it represent?  Answer: 130; the cost to rent the mixer for 3 days

 

 

6. The function T(c) =  predicts the outdoor tem­perature T in degrees Fahrenheit using the number of cricket chirps c per minute. Find T(160). Answer: 80ºF

 

 

 

4. Direct Variation

Text Box: We can describe direct variation by using an equation of the form y = kx

7. The number of purchases n made at a toy store in a mall varies directly with the number of people p entering the mall. Write an equation describing this relationship if 75 purchases were made on a weekday when 5,000 people       mall visited the mall. Answer:

             

8. Use your result from Exercise 7 to find the number of purchases the toy store should make if an attendance of 9,000 is predicted for a special Saturday promotion in the mall.   Answer: 135

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Graphing Using the Rectangular Coordinate System

 

Text Box: CONCEPTS A rectangular coordinate system is composed of a horizontal number line called the x﷓axis and a vertical number line called the y﷓axis. The coordinates of the origin are (0, 0).

 

1. a. Graph the points with coordinates (‑ 1, 3), (0, 1.5), (‑4, ‑4), (2,), and (4, 0).

 

 

x

y

3

-1

0

0

-3

1

 

2. Graph the following sets of coordinates

 

 

 

 

Text Box: To graph ordered pairs means to locate their position on a coordinate system

3. In what quadrant does the point (‑3, ‑4) lie?

 

 

 

 

Equations Containing Two Variables

Text Box: An ordered pair is a solution if, after substituting the values of the ordered pair for the variables in the equation, the result is a true statement.

4. Check to see whether (‑ 3, 5) i s a solution of y =  |2 + x |

 

 

 

 

 

Text Box: Solutions of an equation can be shown in a table of values.

5. a. Complete the table of values and the graph equation y = ‑x3

 

x

y

(x,y)

-2

 

 

-1

 

 

0

 

 

1

 

 

2

 

 

 

 

Text Box: In an equation in x and y, x is called the independent variable, or input, and y is called the dependent variable, or output. To graph an equation in two variables: 1. Make a table of values that contains several solutions written as ordered pairs. 2. Plot each ordered pair. 3. Draw a line or smooth curve through the points.

 

 

 

 

 

 

b. How would the graph of y = ‑x3 + 2 compare to the graph in part 6a.

 

 

 

 

 

 

 

                              Text Box: In many application problems, we encounter equations that contain variables other than x and y.

6. The graph below it shows the relationship between the number of oranges “O” an acre of land will yield if “t” orange trees are planted on it.

 

a. If t = 70, what is “O”?

 

 

b. What importance does the point (40, 18) on the graph have?

 

 

 

 

 

 

 

 

 

 

 

Graphing Linear Equations

 

7. Classify each equation as either linear or nonlinear.Text Box: An equation whose graph is a straight line and whose variables are raised to the first power is called a linear equation.

 

a. y = | x + 2 |                           b. 3x + 4y = 12

c. y = 2x ‑ 3                             d. y =x2 ‑ x

 

 

Text Box: The general or standard form of a linear equation is Ax + By = C where A, B, and C are real numbers and A and B are not both zero.

8. The equation 5x + 2y = 10 is in general form; what are A, B, and C?

 

 

 

9. Complete the table of solutions for the equation 3x + 2y = ‑ 18.

 

x

y

(x, y)

-2

 

 

-8

 

 

 

 

 

 

 

Text Box: To graph a linear equation: 1. Find three (x, y) pairs that satisfy the equation by picking three arbitrary x﷓values and finding their corresponding y﷓values. 2. Plot each ordered pair. 3. Draw a line through the points.

10. Solve the equation x + 2y = 6 for y, find three solutions, and then graph it (see the table in 9 for help).

 

 

 

 

 

 

 

 

Text Box: To find the y﷓intercept of a linear equation, substitute zero for x in the equation of the line and solve for y. To find the x﷓intercept of a linear equation, substitute zero for y in the equation of the line and solve for x. 11. Graph ‑4x + 2y = 8 by finding its x‑ and y‑intercepts.

 

 

 

 

 

 

 

 

 

Rate of Change and the Slope of a Line

Text Box: The slope m of a nonvertical line is a number that measures

12. In each case, find the slope of the line.

a.                 b. 3      c. rise = 5, run = 12,                 d.

 

 

b. The line with the table of values shown here.

x

y

(x,y)

2

-3

(2, -3)

4

-17

(4, -17)

 

 

 

 

 

 

 

 

Text Box:

c. The line passing through the points (2, ‑5) and (5, ‑5)

 

 

 

 

 

Text Box: If P(x1, y1) and Q(x2, y2) are, two points on a nonvertical line, the slope m of line PQ is

d. The line passing through the points –2 (1, ‑4) and (3, ‑7)

 

 

 

 

 

 

 

 

Text Box: Lines that rise from left to right have a positive slope, and lines that fall from left to right have a negative slope, 13. Graph the line that passes through (‑2, 4) and has slope m =

 

 

 

Text Box: If a linear equation is written in slope﷓intercept form, y = mx + b the graph of the equation is a line with slope m and y﷓intercept (0, b).

 14. Find the slope and the y‑intercept of each line.

a. y =  ‑ 2                           b. y = ‑4x

 

 

 

 

15. Find the slope and the y‑intercept of the line determined by 9x ‑ 3, = 15. Then graph it. 

 

 

Writing Linear Equations

Text Box: If a line with slope m passes through the point (x1, y1), the . equation of the line in point﷓slope form is y-y1 = m (x – x1)

16 . Write the equation of a line with the given slope that passes through the given point. Express the result in slope‑intercept form and graph the equation.

 

a.       m = 3, (1, 5)

 

 

 

 

 

b.       m = , (‑4, -1)

 

 

 

 

 

 

 

 

 

17. CAR REGISTRATION When it was 2 years old, the annual registration fee for a Dodge Caravan was $380. When it was 4 years old, the registration fee dropped to $310. If the relationship is linear, write an equation that gives the registration fee f in dollars for the van when it is x years old.

 

 

 

 

 

 

 

 

 

 

 

Functions

Text Box: A function is a rule that assigns to each input value a single output value.

18. In each case, tell whether a function is defined.

 

a. y = 3x – 2                b. Is your age a function of your height?

 

 

 

 

 

Text Box: For a function, the set of all possible values of the independent variable x (the inputs) is called the domain, and the set of all possible values of the dependent variable y (the outputs) is called the range.

19. Find the domain and range of the function.

           

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Text Box: The notation y = f(x) denotes that y is a function of x.

20. For the function g(x) = 1 ‑ 6x, find each value.

           

a. g(l)                                       b. g( ‑ 6)

 

 

c. g(0.5)                                   d. g()

 

 

 

 

 

 

 

 

 

 

 

 

Quiz

Text Box: The graph in Illustration 1 shows the number of dogs being boarded in a kennel over a 3﷓day holiday weekend. Use the graph to answer Problems 1﷓4.

 

 

 

 

 

 

1. How many dogs were in the kennel 2 days before the holiday?

 

 

2. What is the maximum number of dogs that were boarded on the holiday weekend?

 

 

3. When were there 30 dogs in the kennel?

 

 

4. What information does the y‑intercept of the graph give?

 

 

 

5. Graph y = x2 - 4

 

 

 

 

 

 

6. Is (‑3, ‑4) a solution of 3x ‑ 4y = 7?             7. Is y = x3 a linear equation?

 

 

 

 

 

 

8. What are the x‑ and y‑intercepts of the graph of  2x ‑ 3y = 6?

 

 

 

 

 

 

9. Find the slope and the y‑intercept of x + 2y = 8.

 

 

 

 

 

10. What is the slope of the line passing through (‑ 1, 3) and (3, ‑ 1)?

 

 

 

 

 

 

11. What is the slope of a line that is perpendicular to a line with a slope

 

 

 

 

 

 

13. When graphed, are the lines y = 2x + 6 and 6x ‑ 3y = 0 parallel, perpendicular, or neither?

 

 

 

 

 

 

 

 

 

 

                                   

 

14. Is this the graph of a function?

 

 

 

 

 

 

 

 

 

 

15.  Does the equation y = 2x ‑ 8 define a function?                  16. If f(x) 2x ‑ 7, find f(‑3).

 

 

 

 

 

 

 

19. If g(s) = 3.5s3, find g(6).                            

Teamwork

 

 

 

Text Box: DAILY HIGH TEMPERATURE For a 2﷓week period, plot the daily high temperature for your city on a rectangular coordinate system. You can normally find this information in a local newspaper. Label the x﷓axis "observation day" and the y﷓axis "daily high temperature in degrees Fahrenheit." For example, the ordered pair (3, 72) indicates that on day 3 of the observation period, the high temperature was 72°F At the end of the 2﷓week period, see whether any temperature trend is apparent from the graph. Text Box: TRANSLATIONS On a piece of graph paper, sketch the graph of y = | x | with a black marker. Using a different color, sketch the graphs of y = | x | +2 and y = | x | ﷓ 2 on the same coordinate system. On another piece of graph paper, do the same for y = | x | and y = | x + 2 | and y = |x – 2|. Make some observations about how the graph of y = | x | is ,”moved” or "translated" by the addition or subtraction of 2. Use what you have learned to discuss the graphs of y = x2 y =x2 + 2, y = x2 ﷓ 2, y = (x + 2)2 , and y = (x ﷓ 2)2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Text Box: MEASURING SLOPE Use a tape measure (and a level if necessary) to find the slopes of five objects by finding . Record your results in a chart like the one shown in Illustration 1. List the examples in increasing order of magnitude, starting with the smallest slope.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Melissa, has several books on her bookshelf, and if she gets just a few more, she will have over 60 books. Exactly 20% of Melissa's books are math books, and exactly 50% were Christmas presents. How many books does she have on her shelf.

 

 

 

 

 

A man weighs 200 lbs plus one‑fourth of his weight. How much does he weigh?

 

 

 

 

An elephant and a little bird wish to play on a teeter‑totter. Let

E = the weight of the elephant.

b = the weight of the bird.

 

There must be some weight, w (probably very large), so that

 

E = b + w.

 

Multiply both sides by E ‑ b:

E(E ‑ b) = (b + w)(E ‑ b).

 

Using the distributive property:

E2 ‑ Eb = bE + wE ‑ b2 ‑ wb.

 

Subtract ii‑E from both sides:

E2 ‑ Eb ‑ wE = bE – b2 ‑ wb.

 

Use the distributive property again:

E(E ‑ b ‑ w) = b(E ‑ b ‑ w).

 

Divide both sides by E – b - w

E = b.

 

Thus, the weights of the elephant and the bird are the same and they would have no difficulty on the teeter‑totter. Obviously, this reasoning must be false but where is the error?

 

 

 

 

 

Graph each set on it’s own graph paper.

  1. y = x, y = x + 1, y = x – 1
  2. y = 2x, y = 2x + 1, y = 2x – 1
  3. y = x/2, y = x/2 + 1, y = x/2 – 1

 

Discuss the similarities between the graphs and anything else you notice.

 

 

In this exercise you will need to find 4 circular objects.  Use a string and determine the circumference and diameter of the object i.e. use a pop can, table, coffee cup etc…

 

Make graph similar to the one below.  Label your graph like shown, put a scale on your graph too.

 

 

Determine your slope from two non-data points.

 

Helpful Hint: to determine the diameter place the objects between books and measure edge-to-edge of the books.

 

 

What is the slope close to?

 

 

 

 

 

 

In your group discuss the graph in question 14 in your quiz.

  1. Is this a graph of a linear function
  2. If so what is the slope of the function.

 

 

 

Solutions: Solve Linear Equations

 

  1. quadrant III
  2. not a solution
  3. b. It would be 2 units higher
  4. a. 9,000; b. It tells us 40 trees on an acre give the highest yield, 18,000 oranges.
  5. a. nonlinear
  6. A = 5, B = 2, C = 10
  7.  
  8.  
  9. x‑intercept: (‑2, 0); y‑intercept: (0, 4)
  10. b. –7; c. 0; d. 2
  11.  
  12. a. m = ;  y‑intercept: (0, ‑2); b. m ‑4; y‑intercept: (0, 0)
  13. m = 3; y‑intercept: (0, ‑5)
  14. a. y = 3x + 2
  15. f = ‑35x + 450
  16. a. yes; b. no
  17. a. D all reals; R: y £ 0;
  18. a. –5; b. 37; c. –2; d. -8