Lesson 17 Section 3.1 System of Equations (in 2 variables) A System of Equations (in 2 variables) is a set of two equations in two variables for which a common solution is sought. A Solution of such a system is an ordered pair that makes both equations true. I Determining is a given ordered pair is a solution of a System 1. Replace each equation with the correct values from the ordered pair. 2. If both are true, that ordered pair is a solution. Determine if the given ordered pair is a solution of the system shown. 4x y = 9 1) (1, -5) x 3y = 16 2) (-1, -2) x + 3y = 7 3x 2y = 12 3) (3, 1) 3x + 4y = 13 6x + 8y = 26 1
II Solving a System of Equation (in 2 variables) Graphically 1. Graph each equation by an appropriate method you have learned a) Plotting general points b) Plotting the intercepts c) Using slope and y-intercept d) Graphing a vertical or horizontal line 2. Locate the ordered pair where the graphs intersect Solve each system graphically. 3x + y = 5 4) x 2y = 4 (Let s graph these lines by changing the equations to slope-intercept form, then using the slope and y-intercept to graph.) Solution: 2
a b = 1 5) 2a + b = 5 (Let s graph these lines by plotting the intercepts.) a 0 a 0 b 0 b 0 Solution: 3
6) 2x 3y = 6 y = 2 x 2 3 7) If two lines were parallel, how would you describe the solution of the system? If a system of equations in two variables has a solution, we say the system is Consistent. If a system of equation has no solution, we say the system is Inconsistent. If the equations of the lines of a system represent different lines, then the equations are said to be Independent. If the system s equations represent the same line, then the equations are said to be Dependent (really same equation). Here is a summary of the definitions above and what each means graphically. A Lines intersect at one point; system is consistent (solution is one ordered pair) and equations are independent (different lines). 4
B Lines are parallel; system is inconsistent (no solution) and equations are independent (different lines). C Line is the same for each equation; system is consistent (solution is an infinite number of ordered pairs) and equations are dependent (same line). Using a System of Equations of 2 Variables to translate an Application Problem For the lessons over section 1.4, we translated word problems (application problems) to an equation. However, we only used one equation with one variable. Sometimes it is easier or necessary to use two variables. In this case, we must write two equations. For each problem, define two variables and write two equations. (We will not solve these, just write the equations.) 1) The sum of two numbers is 40. The first number is 10 more than the second number. What are the numbers? 1 st number: 2 nd number: 2) In 2002-2003, the average total SAT score for high school students was 1026, with the average math score exceeding the verbal score by 12 points. What were the average math score and average verbal score on the SAT for that year? *Source: College Entrance Examination Board Average math score: Average verbal score: 5
3) Two angles are complementary, have a sum of 90 degrees. The sum of the measures of the first angle and half the second angle is 64 degrees. Find the measures of the angles. 1 st angle: 2 nd angle: 4) Recently an art club purchased 120 stamps for $46. If the stamps were a combination of $0.43 and $0.29 stamps, how many of each type were bought? # $0.43 stamps: #$0.29 stamps: 5) In 2003 Americans bought an average of 76.4 gallons of water and soft drinks per person. The amount of water per person was only 4.3 gallons less that half of the amount of soft drinks per person. How many gallons of each did the average American buy that year? Source: Data from www.beveragemarketing.com Gallons of water: Gallons of soft drinks: 6) A Community College Basketball Squad recently made 40 field goals in a game, some 2-pointers and the rest 3-pointers. These field goals amounted to 89 points. How many of each kind of field goal was made? # 2-pointers: # 3-pointers: 6
7) A Community Center served 250 dinners at a Charity event. A child s plate cost $3.50 and an adult s plate cost $7.00. If the Center raised $1347.50 from these dinners, how many of each type of plate was served? # child plates: # adult plates: 8) Diana furniture outlet regularly sells two popular types of dinette sets; one is in oak and the other in walnut. In July, the outlet sold 5 of the oak sets and 4 of the walnut and collected $18,400 from the sales of these dinette sets. In August, they sold 2 of the oak and 6 of the walnut and collected $17,980. What is the price of each type of dinette set? Price for Oak set: Price for Walnut set: 9) The perimeter of a standard tennis court used for doubles is 228 feet. The width is 42 feet less than the length. Find the dimensions. Width: Length: 7