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BIO 100

BIO 100. Chapter 8 and 9 Variables and Linear Equations. Lecture Week 6— Variables and Linear Equations. By the end of the lecture, students will be able to: 1. Solve problems with one variable and linear equations with two variables.

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BIO 100

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  1. BIO 100 Chapter 8 and 9 Variables and Linear Equations

  2. Lecture Week 6—Variables and Linear Equations • By the end of the lecture, students will be able to: 1. Solve problems with one variable and linear equations with two variables. 2. Explain that the graph of an equation with two variables is the set of all its solutions plotted on a coordinate plane. 3. Interpret key features of graphs and tables in terms of the quantities presented. 4. Sketch graphs showing key features given a description of the relationship between the two quantities. 5. Define what a function is. 6. Define the x-intercept and the y-intercept. 7. Determine the x-intercept and the y-intercept from an equation or a list of points. 8. Differentiate between a positive slope and a negative slope. 9. Find relative maximums and minimums on a graph. 10.Determine symmetries on a graph. 11.Define the domain (with units) for a given function. 12.Calculate the rate of change of a function (presented symbolically or as a table) over a specified interval. 13.Estimate the rate of change from a graph.

  3. Isolating the variable • Solving for an unknown is one of the most important steps in all of mathematics. • It is very much like a balance—each side of the equation must stay equal to the other side. • To ensure that this happens, whatever you do to one side, you must also do to the other.

  4. Isolating the variable (Cont.) • You can add, subtract, multiply, divide, etc. But, whatever you do, be certain the operation is performed on both sides of the equation. • Determine what operation is necessary to “undo” the process performed on the variable. 3x = 6 x – 3 = 7 -2x + 5 = 11

  5. Isolation the variable (Cont.) • Sometimes there will be a variable on both sides. In this case, add the opposite of one of the sides to bring them together on one side of the equation. Example: 2x – 5 = 3x Add -2x to both sides -2x-2x 0 – 5 = x -5 = x

  6. Isolating the variable (Cont.) • The variable must be in the numerator. Example: 12 = 3 x Multiply both sides by x (x) 12 = 3 (x) x 12 = 3x 12 = 3x 3 3 4 = x

  7. Always check your work • After you have isolated the variable, check your work by placing the value you determined back into the original equation. Example: 3x = 6 x = 2 Check: 3x = 6 3(2) = 6 6 = 6 That’s right!

  8. Let’s Practice • Solve the following for the unknown variable. 4x + 7 = 23 8x + 17 = 10x - 1 5x + 30 = 7x 24 + 9 = 19 - x 2x

  9. What is the y-intercept? What is the x-intercept? A point is given by (x,y) The x-intercept is (0,0) The y-intercept is (0,0)

  10. What are the intercepts now?What is the slope? ∆ (delta) is a Greek symbol meaning change. x-intercept is (-2,0) y-intercept is (0,2) ∆x Slope is given by the change in y divided by the change in x ∆y What do you notice about the intercepts? ∆y ∆x

  11. The Phone Bill • Phonetel charges $7 for the first day and $5 each additional day. • What is the constant rate of change for this problem? • Make a table for the first 5 days. ∆x ∆y Now, determine what the values would be at zero days. 0 2 1 1 1 1 5 5 5 5 What does this value tell us about the graph?

  12. The graph The ∆x is 1 (day) The ∆y is 5 (dollars) The slope is $5/1day The y-intercept is (0,2)

  13. Putting it together in an Equation • Slope/Intercept form is given by y = mx + b • m is the slope (or rate of change, like $/day) • b is the y-intercept (just the y-value) • We know the slope is equal to 5, and the y-intercept is (0,2), so what’s the equation for this line? y = mx + b y = 5x + 2

  14. Don’t forget the UNITS • It is always good practice to properly track the units in a problem. Let’s give it a try. Notice how the days cancel to one, and we are left with 5x dollars plus 2 dollars is equal to y dollars. The units check out (dollars = dollars) which means we are likely doing everything correctly.

  15. A Working Table for y = 5x + 2 • Notice that the number of days is out input (x) and the amount of money (dollars) is our output (y).

  16. Solving for just one x-value Sometimes it will be necessary to determine the output (y) for just one x-value. A working table will easily accomplish this, or you may choose to write it a little differently. Given: y = -2x + 6. Determine y when x is 10. When x is 10, y is -14. This means that (10,-14) makes this equation true or is a solution to the equation, and it will be on the graph of this line.

  17. Rates • A rate of change, or slope, will always have a “per”. Can you think of some rates in our daily lives? • An important aspect of this that the rate must always cancel out the units of the input and leave you with the units of the output. Example: Which is the rate? Input? Output? This equation would give us the distance driven in two hours while driving at 50 miles/hour.

  18. The Domain and Range • The domain is all of the possible inputs (x-values) for a situation or equation. • The range is all of the possible outputs (y-values) for a situation or equation. • Many times the domain and range will be all real numbers (R). But there are many exceptions. The domain for y = 1/x must never include 0 because you may never divide by zero. The domain would be all real number except for 0.

  19. The Domain and Range (Cont.) • Most of the time in science, the domain and range will be determined by the context of the problem. Example: A person was driving at 50 miles/hour. How far had she driven in 3 hours? This example involves a person, and people have limits. You would expect that the domain would only include positive times and be limited to around 40 hours. Domain: 0<x<40 (hours) Range: 0<y<2,000 (miles)

  20. The Domain and Range (Cont.) • The mathematical equation that models this real-world problem is y = 50x or, with units: y (miles) = 50 (miles/hour) x (hours) The mathematical equation would have a domain and range of all real numbers, but the real-world problem does not share these values.

  21. Functions • A function is a process that will have exactly one output for every input. This means that you cannot put 5 into the function machine one time and get 10, and then put 5 in again and get something different than 10—you must always get the same output for a given input. • The function notation is written as f(x), which means that you take the input of “x” and perform the function on it. This is said “f of x”

  22. Functions (Cont.) • Example: y = 3x + 2 slope/intercept form f(x) = 3x + 2 function notation Find f(4). This is the same problem as “find y when x is 4.” Said “f of 4” This is function notation

  23. Finding out more • Given: slope is 3 and a point on the line is (2,4) • Find another point on the line • Find the equation of the line in y = mx+b form • Given: two points on a line are (1,3) and (2,5) • Find the slope • Find the equation for the line in y=mx+b form Here’s one more: (3,7) y = 3x + -2 slope is 2 y = 2x + 1

  24. Finding out more . . . • Using algebra, answer these questions: • Given: y=3x + 5 What is y when x is 10? • Given: y=-2x + 6 What is x when y is 4? • Given: y=(1/3)x + 5 What is y when x is 9? y = 35 y = -2x + 6 4 = -2x + 6 -2 = -2x 1 = x y = (1/3)x + 5 y = (1/3)(9) + 5 y = 3 + 5 y = 8

  25. Work Time Try these problems on for size! 1. Solve for the variable: 3x = 24 -4x – 5 = 15 20 – x = 25 2. Solve this problem using 2 different methods—use a table and a graph. • You are deciding which phone company to go with. Phonetel charges $7 for the first day and $5 each additional day. Celltech charges $15 for the first day and $4 each additional day. • Which is the better deal? • For what amount of days is Phontel better? Celltech better? Is there a point of intersection (where they are both equal)?

  26. Exit Quiz and Homework • Exit Quiz—Copy the questions, then answer. Place your name/date/class day & time in the upper right hand corner. 1. Given: y = -3x + 10 What is the slope? What is the y-intercept? Make a table for the whole number x-values, 0 to 4. Graph it. 2. Solve for x 16 = 2 5x – 6 = -4x + 12 x Homework • Read and annotate Chapters 8 and 9. • Math Project is due • Review your notes, the syllabus and course objectives from class. (Be sure you understand the objectives.)

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