ALGEBRA II

track 2





CHAPTER 3



Section 1-Open Sentences in Two Variables

This section introduces the idea of 2 variables in a problem at the same time. We can only solve with one variable. This section deals with solving by substituting a given value in for a variable and then solving for the remaining variable.

The second portion of this section deals with finding your own numbers to plug in for the variables. In both regular problems and word problems, by using the given information it is possible to limit the possible choices for a variable. The key to these problems, however, is to give all possible answers.


Section 2-Graphs of Linear Equations in Two Variables

Going back to the beginning of section one and the infinite possible answers, we know learn to show this graphically as it is impossible to list all the possible answers. Like section 1, pick an x value, plug into the equation and get the y value that goes with it. What is different here is that you can choose any x value that you want. You must find 3 ordered pairs to make sure you have the proper answer. Ordered Pairs are written (x,y) with x being the horizontal movement and y the vertical movement on the graph. Plot the 3 points and connect them using a straight edge and it should be a line. If not, go back and verify your answers for your 3 points and check signs, operations etc..


Section 3-The Slope of a Line

This section allows you to find the slope, or steepness, of a line in 2 different ways. The first method you need to know points that are on the line, whether they are given to you or you can find them from the graph of the line. Once you have the two points, plug them into the formula. It doesn't matter which point you consider point one and two. What is important is which ever y value you put first, its x value goes first on the bottom. The other method is used if you are given the equation of the line, in the form Ax+By=C. The slope is found by simply doing -A/B. Of course, you can pick 2 x values, sub in and get the y values like section 2 and then use the slope formula.

examples:

given points (3,4) and (5,1) you can do either (4-1)/(3-5) or (1-4)/(5-3) either way the slope is -3/2
given equation 3x-2y=6 the slope is - (3/-2) or 3/2

Section 4-Finding the Equation of a Line

This section is basically the reverse of the last 2. Now you know information about the graph or specific points of the line and you want to find the equation of the line. There are 2 forms to use to start the process. The first is called Point-Slope Form and the second is called Slope-Intercept Form . In point slope form you need to know one point and the slope of the line. Plug the info into the proper place in the formula and solve. Slope intercept (y = mx + b) is a special case of point slope and can only be used when the given point is the y-intercept. Again plug the info in the correct place and solve. Unless told otherwise, the final answer should be in Standard Form (Ax + By = C) .

Using the above example we found the slope to be -3/2. Now take either point and plug in and we get y-4 = -3/2(x-3) or y-1 = -3/2(x-5) mult both sides by 2 to get rid of the fraction and put proper form and you get 3x+2y = 17 for the equation.

Section 5-Systems of Linear Equations in Two Variables

This section deals with solving 2 equations at a time. You want to find the ordered pair that works both equations at the same time. As we saw in an earlier section, this can be done graphically, but that is very inaccurate, especially if the answer involves fractions. When you solve a system of equations there are only 3 possible solutions: one answer, no answer or infinite answers. One answer occurs when the lines intersect and this is the most common case. No answer occurs when the lines are parallel. Infinite answer occurs when the lines coincide, or lay on top of each other. There are 2 ways to solve these systems: Substitution and addition methods.
Substitution Method
step 1. Solve one of the equations for one of the variables.
2. Substitute this answer into the other equation.
3. Solve this new equation for the remaining variable.
4. Take the answer to step 3 and put back into either original equation.
5. Solve for the remaining variable.
6. Check answers in both equations to make sure they work.
Addition Method
1. Make one set of coefficients opposites of each other by multiplying one of both equations by a number.
2. Add these 2 new equations together. If both variables are still in the new result, then step 1 was not done properly.
3-6 Same as the substitution method listed above.

Examples: 3x + y = 6 solve this equation for y so y = -3x + 6
              2x + 3y = 11 Substituting in gives us 2x + 3(-3x + 6) = 11
Distributing and combining like terms gives up -7x + 18 = 11 and then -7x = -7 and x = 1
putting the x = 1 into the top equation gives up 3(1) + y = 6 and thus y = 3. Checking this in the second equation show that 2(1) + 3(3) does indeed equal 11 so it works both equations.

To do this same system by the addition method you can multiply the top equation by -3 and get -9x - 3y = -18
When you add that to the second equation the y's cancel out and you are left with -7x = -7 which is exactly what you got after the substitution step above. The remainder of the problem is done as above. You could also multiply the first equation by 2 and the second by -3 to get the x's to cancel out.

Keep in mind there is more than one way to solve the system of equations but only one final answer.

Section 6-Problem Solving:Using Equations
This section applies the information from section 5 to solve word problems. Many of the problems from chapter 1 can now be set up using two variables instead of one, especially coin problems. The new type of problem involved the distance formula. Now instead of just one part to the speed, or rate, there are now 2 parts. All these problems dealt with airplanes but also could be applied to boats in water. There are 2 types of wind to consider. Head Wind blows into you and slows you down so you subtract that from your normal speed. Tail Wind blows from behind and thus helps speed you up. Therefore you add this to your normal speed to get your new speed. Fill in the chart the same as before with this new setup.

Example: With a tail a plane travels 300 miles in 1 hour 30 minutes. Returning against the same wind took 2 hours. Find the air speed of the plane and the speed of the wind.
Rate Time Distance
p+w3/2 300
p-w 2 300

using rate x times = distance gives us 3/2(p+w)=300 and 2(p-w)=300 Multiply first equation by 2/3 and divide second by 2 gives us p+w=200 and p-w=150 solving as above in section 5 p(plane speed) = 175 and w (wind speed) = 25

Section 7-Linear Inequalities in Two Variables
Treat the inequality problem like the equations from the beginning of the chapter. You can "solve" by picking x values and finding the y that goes with them or put in slope intercept form to graph (I think you will find this the easier way overall). When it comes time to graph, there are 2 additional steps to do: 1) determine whether the line will be solid or dotted. You can determine this by the type of inequality. If the equal is part of the problem use a solid line, if just a < or > use dotted to show the line is not part of the answer (similar to the open and closed circles on a number line). 2) Shade the area above the line if >, below the line if < . You can double check this by picking a point and substituting into the inequality. If you get a true statement you shade that area. If its a false statement, shade the other area.

Section 8-Functions
Section 9-Linear Functions
Section 10-Relations