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**Unit 1 Modeling with Linear Equations**

Learning Goal: I can model real-life situations using linear equations

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**Warm-Up: Graph of the Day**

What are the y-intercepts for the yellow cab and the blue cab? What do these values represent. You miss the bus and need to take a cab 5km to school; which cab should you choose? If you wanted to travel as far as possible with $6, which cab should you choose?

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Linear Systems Many real-life situations can be modeled using linear equations. When 2 or more equations are used, it is called a system of equations. If all of the equations are linear, it’s called a linear system.

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Linear Systems The solution to a linear system is an ordered pair that satisfies all equations Graphically, the solution is the point where two lines intersect

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**Modeling with Linear Equations**

At the zoo gift shop, posters of polar bears are $1 and poster of penguins are $6. Charles spent $32 to buy 7 posters, how many of each did he buy? Let x represent the number of polar bear posters Let y represent the number of penguin posters You were probably introduced to these kinds of problems in elementary school; they may have been called ‘guess and check’ problems or ‘trial and error’ problems. These problems can be represented with linear systems During this unit, we will learn a more prescribed method for solving these problems (more accurate than guess and check). One of the most challenging aspects is constructing the equations based on a written description. Today we are going to practice constructing linear systems and not worry about solving the problem just yet. In these problems, there are usually 2 types of information (in this case # and cost). We need one equation for each type of info. Start by defining variables (use the question ‘how many’ as a guide) and use variable to come up with 2 equations (one for number and one for cost).

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**Modeling with Linear Equations**

Jamie went out to her grandfather's farm. Her grandfather has pigs and chickens on his farm. She noticed that there were a total of 26 heads and 68 feet among them. How many chickens and how many pigs did her grandfather have? What are the 2 pieces of information? Define variables Use variables to make one equation for each type of information

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**Modeling with Linear Equations**

Yasmin has $8000 to invest and would like to earn $500 from the money. How much should she invest in a stock that has been paying 10% annual return and how much should she invest in savings bonds that pay 4% annual interest? What are the 2 pieces of information? Define variables Use variables to make one equation for each type of information

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**Modeling with Linear Systems**

José travelled 95km from Oakville to Oshawa by car and GO train. The car averaged 60km/h, and the train averaged 90km/h. The whole trip took 1.5h. How long was he in the car? In the train? What are the 2 pieces of information? Define variables Use variables to make one equation for each type of information

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Homework Pg # 7, 10, 19, 23, 26 Just define variables and write 2 equations; you don’t need to solve.

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Solving Problems using Linear Systems. Type #1: Arithmetic The sum of two numbers is 255. When the smaller is subtracted from the larger, the result is.

Solving Problems using Linear Systems. Type #1: Arithmetic The sum of two numbers is 255. When the smaller is subtracted from the larger, the result is.

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