Biology 198

Chapter 8 and 9 Variables and Linear EquationsBIO 100

Lecture Week 6Variables and Linear EquationsBy 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.

2Isolating the variableSolving for an unknown is one of the most important steps in all of mathematics.It is very much like a balanceeach 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.

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 = 6x 3 = 7-2x + 5 = 113x = 6 Multiplication is UNDONE by division; divide both sides by 3.It is best to view the second one as add -3 (subtraction is adding the opposite, so re-write it as x + -3 = 7), so to undo that we would add +3 to both sides.If we say the third one as multiply a number by -2 then add 5, we see that we should add -5 then divide by -2. (Feel free to draw a process diagram on the board.)4Isolation 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 = 3xAdd -2x to both sides-2x -2x 0 5 = x -5 = xGenerally speaking, it is easier to choose the smaller of the two to move. However, either way will result in the same answer.

You can write the -2x under the 3x OR you can write it all on one line, like:2x 2x 5 = 3x 2x

Show students BOTH ways and explain that they are equivalent.5Isolating the variable (Cont.)The variable must be in the numerator.Example: 12 = 3 xMultiply both sides by x (x) 12 = 3 (x) x 12 = 3x 12 = 3x 3 34 = xNotice that the answer is written with the x on the right-hand side of the equation. Is that OK? (Yes. You can always flip it if you want to.)6Always check your workAfter 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 = 63(2) = 66 = 6Thats right!Always check your work.7Lets PracticeSolve the following for the unknown variable.

4x + 7 = 23 8x + 17 = 10x - 1

5x + 30 = 7x 24 + 9 = 19 - x 2x

Always check your work 8What 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)The x-intercept is the point at which the graph crosses the x-axis. And the y-intercept is the point at which the graph crosses the y-axis. How is a point labeled on a co-ordinate grid? (x,y) What is the x-intercept? (0,0) What is the y-intercept? (0,0) The graph crosses both axes at the same point.9What are the intercepts now?What is the slope?

x-intercept is (-2,0)y-intercept is (0,2) y xSlope is given bythe change in y divided by the change in xyxWhat do younotice about theintercepts? (delta) is a Greek symbol meaningchange.What do you notice about the y-value of the x-intercept? (It is always zero.) What do you notice about the x-value of the y-intercept? (It is always zero.)

The delta y is one . . . you increase one unit on the y-axis as you move from 2 to 3. The delta x is also one . . . you increase one unit on the x-axis as you move from 0 to 1. So the slope is 1/1 or 1.

10The Phone BillPhonetel 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 (days)y (dollars)17212317422527yx11115555Now, determine what the values would be at zero days.02What does this value tell us about the graph?Notice that the (delta y)/(delta x) is 5. We can see this already from the problem.

The (0,2) value tells us the y-intercept. When we graph this, the line will cross the y-axis at (0,2).11The graph

The y-interceptis (0,2)The y is5 (dollars)The x is 1 (day)The slope is $5/1dayPutting it together in an EquationSlope/Intercept form is given by y = mx + bm 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 whats the equation for this line?y = mx + b

y = 5x + 2Dont forget the UNITSIt is always good practice to properly track the units in a problem. Lets 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.

14A Working Table for y = 5x + 2Notice that the number of days is out input (x) and the amount of money (dollars) is our output (y).

x (days)5x + 2y (dollars)05(0) + 2215(1) + 2725(2) + 21235(3) + 21745(4) + 22255(5) + 227x5(x) + 2yWhenever you substitute a number for a variable (x), always use parenthesis. This will avoid many problems.

This is a functionevery input (x) has exactly one corresponding output (y).15Solving for just one x-valueSometimes 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.x=10y = -2x + 6y = -2(10) + 6y = -20 + 6y = -14When x is 10, y is -14.This means that (10,-14) makesthis equation true or is a solution to the equation, and it will be on the graph of this line.(Last thing said before leaving this slide . . .) How many solutions are there to this equation? (Infinite)How many points are there on the graph of this line? (Infinite)16RatesA 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.

The Domain and RangeThe 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.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