STUDY GUIDE FOR TEST 2. Project 3: Fill in the blanks and do the assigned questions. 11/6/07

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1 STUDY GUIDE FOR TEST 2 Name: Project 3: Fill in the blanks and do the assigned questions. 11/6/07 Quadrant II Quadrant I ORDERED PAIR: The first number in the ordered pair is the -coordinate and the second number in the ordered pair is the -coordinate. (, ) -axis Origin (, ) Quadrant III Quadrant IV Linear Equations Slope = change in change in Lines l 1 and l 2 are to each other. The slope of l 1 is the as the slope of l 2. Lines l 1 and l 2 are to l 3. The slope of l 1 is the of l 3. Slope of l 1 = -axis y-intercept of l 1 is (, ) To find the y-intercept, set =0 and solve for y l 3 x-intercept of l 1 is (, ) To find the x-intercept, l 2 set =0 and solve for x l 1 Equation for l 1 is y= Equation for l 2 is y = Equation for l 3 is y = Given two points on a line (x 1,y 1 ) and (x 2,y 2 ) The slope can be found through the equation m = - - The slope-intercept form of a linear equation is y =, where m is the slope and ( 0,b ) is the y-intercept. The point slope formula for a line with point (x 1, y 1 ) and slope m is = ( ) Slope of l 3 = Slope of l 2 = Equation for this horizontal line is This line has a slope of Equation for this vertical line is This line has NO SLOPE

LINEAR INEQUALITIES To graph a LINEAR INEQUALITY, First rewrite the inequality to solve for y. If the resulting inequality is y >., Then make a dashed line and shade the area the line.

2 LINEAR INEQUALITIES To graph a LINEAR INEQUALITY, First rewrite the inequality to solve for y. If the resulting inequality is y >., Then make a dashed line and shade the area the line. If the resulting inequality is y <.., Then make a dashed line and shade the area the line. If the resulting inequality is y.., Then make a line and shade the area the line. If the resulting inequality is y., Then make a line and shade the area the line. Graph the solution set of 3x 2y 12

3 Input x FUNCTIONS Input 3 Function f(x) Output y=f(x) Function f(x)=2x-1 Output f(3) = A function, f, is like a machine that receives as input a number, x, from the domain, manipulates it, and outputs the value, y. The function is simply the process that x goes through to become y. This machine has 2 restrictions: 1. It only accepts numbers from the domain of the function. 2. For each input, there is exactly one output (which may be repeated for different inputs). OFFICIAL DEFINITION OF A FUNCTION: Let x and Y be two nonempty sets. A function from x into Y is a relation that associates with each element of X, exactly element of Y. However, an element of Y may have more than one elements of x associated with it. That is, for each ordered pair (x,y), there is exactly y value for each x, but there may be multiple -values for each y. The variable x is called the independent variable (also sometimes called the argument of the function), and the variable y is called dependent variable (also sometimes called the image of the function.) Analogy: In the x-y relation -ship, the x s are the wives and the y s are the husbands. A husband is allowed to have more than one wife, but each wife(x) is only allowed 1 husband(y). A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, then x corresponds to y, or y depends on x. x Hours Studying in Math Lab y Score on Math Test The set of x-coordinates {2,3,4,4,5,6,6,7} corresponds to the set of y coordinates {60,70,70,80,85,85,95,90} The set of distinct x-coordinates is called the of the relation. This is the set of all possible x values specified for a given relation. The set of all distinct y values corresponding to the x- coordinates is called the. In the example above, Domain = {2,3,4,5,6,7} Range = {60,70,80,85,85,95,90} This relation is not a function because there are two different y-coordinates for the x-coordinate, 4, and also for the x-coordinate, 6.

4 WORD PROBLEMS Simple Interest Interest earned ($$) = Principal * Rate * time (initial investment) * (interest rate )* (1 year for annual interest) Mixture Problems Quantity of a Substance in a solution = % of concentration * Amount of Solution Example: Forty ounces of a 60% gold alloy means that the quantity of gold in the alloy is.60 * 40 = 24 ounces of gold. Distance Problems Distance = rate * time If distance is in miles, and rate is in mph, then time must be in If time is in minutes, then multiply time by --- to convert to hours. Setting up word problems: 1) Find out what you are being asked to find. Set a variable to this unknown quantity. Make sure you know the units of this unknown (miles?, hours? ounces?) 2) If there is another unknown quantity, use the given information to put that unknown quantity in terms of the variable you have chosen. (For example, if total distance traveled is 700 miles, then part of the trip is x miles and the other part of the trip is 700 x miles.) 3) Set up a table with a row for each unknown and columns made up of the terms of one of equations above (r*t = d, Pr = I, etc..) 4) Use the given information to combine the equations of each row of the table into one equation with one variable to solve for. 5) Once one variable is solved for, you can find the other unknown. (For example, is x = 100 miles, then the other part of the trip is = 600 miles.) 6) Check your equation by plugging in your value for x and seeing if your equation is true.

5 SYSTEMS OF LINEAR EQUATIONS A system of linear equations is a set of two equations of lines. A solution of a system of linear equations is the set of ordered pairs that makes each equation true. That is, the set of ordered pairs where the two lines intersect. If the system is, there is ONE SOLUTION, an ordered pair (x,y) If the system is, there is INFINITELY MANY SOLUTIONS, all (x,y) s that make either equation true (since both equations are essentially the same in this case. If the system is, there are NO SOLUTIONS, because the two equations represent parallel lines, which never intersect. GRAPHING METHOD. Graph each line. This is easily done be putting them in slope-intercept form, y = mx + b. The solution is the point where the two lines intersect. SUBSTITUTION METHOD. 2x y = 5 3x + y = 5 Choose equation to isolate a variable to solve for. In this system, solving for y in the second equation makes the most sense, since y is already positive and has a coefficient of 1. This second equation turns into y = -3x + 5 Now that you have an equation for y in terms of x, substitute that equation for y in the first equation in your system. Substitute y = -3x + 5 in 2x y =5 2x (-3x + 5) = 5 Simplify and solve for x. 2x + 3x 5 = 5 5x 5 = 5 5x = 10 x = 2 ADDITION METHOD 5x + 2y = -9 12x 7y = 2 Eliminate one variable by finding the LCM of the coefficients, then multiply both sides of the equations by whatever it takes to get the LCM in one equation and LCM in the other equation. After this we can add both equations together and eliminate a variable. Let s choose to eliminate y. The y terms are 2y and -7y. The LCM is 14. Multiply the first equation by 7 and the second equation by 2. 7(5x + 2y) = -9(7) 2(12x 7y) = 2(2) 35x + 14y = x 14y = 4 Add them together 59x = -59 x = -1 Solve for y by substituting x=2 into y = -3x + 5. y = -3(2) + 5 = = -1 Therefore solution is (2,-1) CHECK by substituting solution into the other equation and see if it is true. 3x + y = 5 3(2) + (-1) =5 5 = 5 YES! Solve for y by substituting x=-1 into either of the two original equations. 5(-1) + 2y = -9 2y = -4 y = -2 Therefore solution is (-1,-2) CHECK by substituting solution into the other equation and see if it is true. 12 x 7y = 2 12(-1) 7(-2) = = 2 2 =2 YES!

6 REVIEW QUESTIONS p #7,11,13 p #1-25 odd p. 365 #1-23 odd

CHAPTER 5 LINEAR SYSTEMS

CHAPTER 5 LINEAR SYSTEMS Systems of Linear equations have either one solution (independent), no solutions (inconsistent), or infinitely many solutions (dependent). An independent system is the case when