Unit 7: It s in the System

Transcription

1 Unit 7: It s in the System Investigation 1: Linear Equations with Two Variables I can convert between standard and slope intercept forms, and graph systems of equations. Solving equations is one of the most common and useful tasks in mathematics. In earlier Units, you learned how to solve: Linear equations, such as 3x + 5 = 17 Proportions, such as 3 = 6 5 x Quadratic equations, such as x 2 5x + 6 = 0 Exponential equations, such as 2 x+1 = 64 The problems in this investigation pose a new challenge. You will learn to solve linear equations, such as 3x + 5y = 11, that have two variables. What does a solution for a linear equation look like? What does a graph for a linear equation look like?

2 Investigation 1 Lesson 1: Solving Equations with Two Variables The third years are selling T-shirts and caps to raise money for their end-of-year party. The profit from the fundraiser depends on the number of caps and the number of T-shirts sold. Two numbers that fit both the x-value and y-value of a linear equation are called a. We write for linear equations as After this lesson you should be able to answer the question: How can we solve equations with two variables? Problem 1.1 To plan for the fundraiser, class officers need to know how many T-shirts and caps to order and sell. A. Find the profit P if the students sell shirts and 10 caps

3 2. 12 shirts and 20 caps 3. s shirts and c caps B. 1. Find five pairs of numbers for shirt and cap sales that will allow the students to make a profit of exactly $ Each answer from part (1) can be written as an ordered pair of numbers (s, c). The ordered pairs (s, c), which represent points on a graph, are solutions of the equation 5x + 10c = 600. Plot the ordered pairs on the coordinate grid below. 3. Use the graph to find three other ordered pairs that meet the profit goal.

4 C. For each equation below Find FIVE solution pairs (x, y), including some with negative values Plot the solutions on a coordinate grid and draw the graph showing all possible solutions (draw the line that represents the equation) 1. x + y = x 2y = -4 Practice ACE Questions: # 1 & 2 Investigation 1 Lesson 2: Standard to Slope-Intercept Form There are two common forms of linear equations with two variables. When the values of one variable depend on those of another, it is common to express the relationship as. This equation is in form.

5 When the values of the two variables combine to produce a fixed third quantity, you can express the relationship as. This equation is in form. The graph of each type of equation is a straight line. Therefore, it is natural to ask: Given an equation in one form, how can you rewrite the equation in the other form? The following steps will walk you through the process of writing equations from standard form to slope-intercept form: STEP 1: EXAMPLE: STEP 2: STEP 3: After this lesson you should be able to answer the question: How can we rewrite linear equations from standard form to slope-intercept form?

6 Problem 1.2 A. Four students tried to write 12x + 3y = 9 in y = mx + b form. Did each student successfully rewrite the equation? If not, tell what errors the student made. a. Jeri s equation: N = s + s + s + s + 4 b. Hank s equation: N = 4(s + 2) c. Sal s equation: N = 2s + 2(s + 2) B. Write each equation in y = mx + b form. 1. 8x + 4y = x - y = 9 C. Write each equation in standard (Ax + By = C) form. 1. y = 5 3x 2. x = 2y - 3

7 D. Write a linear equation in either slope-intercept or standard form represent each situation. Then, explain why your choice is the best representation. 1. Mary is selling popcorn for $5.00 per bucket and hotdogs for $4.75 each. After one hour, her total was $ Matt is in charge of selling roses for the Valentine s Day dance. The roses sell for $3.75 each. He estimates that the expenses for the roses are $ He wants to write an equation for the profit. 3. Karlee is mixing paint for an art project. She mixes 5 ounces of green paint with every 3 ounces of white paint. She needs 50 ounces of the paint mixture total. Practice ACE Questions: 3-9 Investigation 1 Lesson 3: Intersecting Lines. At a school band concert, Christopher and Steven sell memberships for a band s booster club. An adult membership costs $10, and a student membership costs $5. At the end of the evening, the students had sold 50 memberships for a total of $400. The club president asked: How many of the new members are adults and how many are students? You can answer the question by that represent the question and the given information.

8 Problem 1.3 A. Let a represent the number of adult memberships (that cost $10 each) and s represent the number of student memberships (that cost $5 each). 1. Write an equation that relates a and s to the $400 total income. 2. Find three solutions for your equation from part (1). 3. Write and equation that relates a and s to the total of 50 new members? 4. Find three solutions for your equation from part (3). 5. Are there any pairs of values for a and s that are solutions for BOTH equations? B. 1. Graph the two equations you used in part (a) on the grid. 2. What are the coordinates of the intersection point? 3. Check your answer by substituting the x and y values in for a and s in BOTH equations and make sure they work!

9 The two equations you wrote to model the conditions of this problem are called a. The coordinate of the intersection point work for both equations at once. These coordinates are called the. There is for a system of linear of equations. It is helpful to find the - intercept and - intercept to help you create your graph. C. Use a graph to solve each system of linear equations (try using the intercepts). CHECK YOUR ANSWER in both equations! 1. x + y = 4 and x y = x + y = -1 and x 2y = 7 Practice ACE Questions: #10-13