Using Linear Equations

Today’s lesson will focus on real life situations where you’ll need to use math. We’ll look at the information we’re given, set up equations, solve them and use the results to make decisions. We’ll keep the solving methods simple today but learn more sophisticated methods in later lessons. Long after you have completed my class, you will use these skills to help you in your personal finances and life’s decision making. Content Objective: The student will identify real life situations where math can be used. The student will solve systems of linear equations The student will draw conclusions from the results of their calculations Learning Objective: The student will observe situations involving math and retell it using mathematical symbols The student will define and label needed variables The student will write systems of linear equations Standards: A.CED.1: Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. A.CED.2: Create equations in two or more variables to represent relationships between quantities. A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Math in Real Life Watch this video about how math is used in real life. As you watch, jot down in your notes some of the occupations you hear mentioned that use math. What occupations can you derive based on the photos and your knowledge of different jobs. What activities and situations are mentioned that you would use math for?

Where/when do you see math used? Occupations / jobsSituations / activities When else might you use math?

What are linear equations? A linear equation is an equation for a straight line (hence “line” in its name). It is made up of 2 or more expressions joined with an equals symbol (“=“). The expressions can also be joined with a greater than or less than symbol (“>” or “<“); then we call it a linear inequality. When we look at more than one linear equation at a time to solve a problem, it’s called a system of linear equations. What can we use linear equations for?

Are these linear equations? Why or why not? X + 4 = 7 X + 4 = y X 2 + 2x = 9 No, it’s a single point. But it can be a necessary part for solving a system of equations Yes, it forms a line and there is a unique answer, y, for each x No, it has an exponent; that makes it non-linear

How to set up linear equations This is very similar to our previous lesson on translating word problems Write a verbal model Write down the words for each term Add the math symbols to connect them Assign labels Determine which terms are constants and which can change (variables) Choose a letter for each variable Write algebraic model Substitute the labels in your verbal model Simplify where possible

How to solve a system of linear equations Graphing Substitution A useful method for approximating a solution, checking the reasonableness of a solution, and providing a visual model. 1. Write each equation in a form that is easy to graph 2. Graph both equations in the same coordinate plane 3. Estimate the coordinates of the point of intersection 4. Check the coordinates algebraically by substituting into each equation of the original linear system A useful method when one of the variables has a coefficient of 1 or Solve one of the equations for one of its variables 2. Substitute the expression from Step 1 into the other equation and solve for the other variable 3. Substitute the value from Step 2 into the revised equation from Step 1 and solve 4. Check the solution in each of the original equations The simplest solution methods (and the only ones we will use for now) are graphing and substitution

Redbox or Netflix? Setting it up Write Verbal model # of movies * cost per movie = Redbox cost membership fee = Netflix cost Assign labels # of movies watched/month = M Cost per movie = $1.20 Membership fee = $7.99 Cost of getting movies from Redbox = R Cost of getting movies from Netflix = N Write algebraic model 1.2M = R (eq. 1) 7.99 = N (eq. 2) Let’s solve by graphing Since the lines cross between 6 and 7 movies per month, Netflix is a better deal if you watch 7 or more movies per month You and your friends like to get together to watch movies. Usually, you just go to the local Redbox to pick something out. But you’re thinking about joining Netflix instead. Movies at Redbox are $1.20 each; Neflix charges $7.99/month for unlimited movies. How many movies would you have to watch each month for Netflix to be the better deal? What other considerations (not necessarily math related) might influence your decision?

An Easy “A” Work with your partner to set up the system of equations S = softballs; they cost $2.75 each B = baseballs; they cost $3.25 each There were a total of 80 balls ordered at a total cost of $ S B = 245 (eq. 1) S + B = 80 (eq. 2) Let’s solve by substitution If S + B = 80, then B = 80 – S Substituting B into eq. 1: 2.75S (80 – S) = 245 Simplifying: 2.75S – 3.25S = = 0.5S 30 = S Substituting S back into eq B = 80 B = 50 Therefore, there were 30 softballs and 50 baseballs ordered You’ve got an easy class this semester: you’re the aide for the P.E. coach. Unfortunately, while putting away some papers, you spilled your soda on coach’s desk and the ink smeared on the order for baseballs and softballs. What to do?!? You can see that there were a total of 80 balls purchased, the softballs are priced at $2.75 each, the baseballs cost $3.25 each, and the total bill is $245. Can you figure out how many of each ball was ordered? What’s the best method to solve this? Try it with your partner and we’ll check our results

What to buy? It’s time to shop for school clothes. Your favorite store is running a sale: all bottoms are $20 each and all tops are $10 each. Write the equation for how many tops and bottoms you can buy. What other information do you need to decide what to buy? A budget. Mom gave you $200 What does your system of equations look like now? You and your partner each solve it. Do you get the same answer? What happened? Let B= # of bottoms, T= # of tops and C=total cost 20B + 10T = C C = 200

You got a job offer! You are offered two different sales jobs. Job A offers an annual salary of $30,000 plut a year-end bonus of 1% of your total sales. Job B offers an annual salary of $24,000 plus a year-end bonus of 2% of your total sales. a. How much would you have to sell to earn the same amount in each job? b. You believe you can sell between $500,000 and $800,000 of merchandise per year. Which job should you choose? Solve this on your own then compare your answer with your neighbor. Explain your solution method. What other factors might you consider when deciding which job to take?

Tonight’s Homework Page 421 – 423 Problems: 13 – 15, 19 – 45 every other odd, 46, 48,

Resources Images Content Shopping teens on chart 11: content/uploads/2013/06/online-fashion-shopping.jpg Video on chart 3: