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I can explain the solution to a system of equations in a real-world context.
I can explain what a system of equations is.
I can make graphs to find an ordered pair that two real-world situations have in common.

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passing on the trail

There is a hiking trail near the town where Han and Jada live that starts at a parking lot and ends at a lake. Han and Jada both decide to hike from the parking lot to the lake and back, but they start their hikes at different times.

At the time that Han reaches the lake and starts to turn back, Jada is 0.6 miles away from the parking lot and hiking at a constant speed of 3.2 miles per hour towards the lake. Han’s distance, d, from the parking lot can be expressed as d = -2.4t + 4.8, where t represents the time in hours since he left the lake.

1. What is an equation for Jada’s distance from the parking lot as she heads toward the lake?

2. Draw both graphs: one representing Han's equation and one representing Jada’s equation. It is important to be very precise! Be careful, work in pencil, and use a ruler.

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passing on the trail, cont.

Applet

Find the point where the two graphs intersect each other. What are the coordinates of this point?
What do the coordinates mean in this situation?
What has to be true about the relationship between these coordinates and Jada’s equation?
What has to be true about the relationship between these coordinates and Han’s equation?

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stacks of cups

A stack of small cups has a height, h, in centimeters of h = 1.5n + 6. A stack of n large cups has a height, h, in centimeters of h = 1.5n + 9.

Graph the equations for each cup on the same set of axes. Make sure to label the axes and decide on an appropriate scale.

applet

For what number of cups will the two stacks have the same height?

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Lesson Summary

A system of equations is a set of 2 (or more) equations where the variables represent the same unknown values. For example, suppose that two different kinds of bamboo are planted at the same time. Plant A starts at 6 ft tall and grows at a constant rate of ¼ foot each day. Plant B starts at 3 ft tall and grows at a constant rate of ½ foot each day. We can write equations

y = ¼ x + 6 for Plant A and y = ½ x + 3 for Plant B, where x represents the number of days after being planted, and represents height. We can write this system of equations.

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Lesson Summary, Cont.

Solving a system of equations means to find the values of x and that make both equations true at the same time. One way we have seen to find the solution to a system of equations is to graph both lines and find the intersection point. The intersection point represents the pair of x and y values that make both equations true. Here is a graph for the bamboo example:

The solution to this system of equations is (12,9), which means that both bamboo plants will be 9 feet tall after 12 days.

We have seen systems of equations that have no solutions, one solution, and infinitely many solutions.

when the lines do not intersect, there is no solution. (parallel lines)
When the lines intersect once, there is one solution
When the lines are right on top of each other, there are infinitely many solutions.

*Some systems can’t be easily solved by graphing, but can be easily solved using algebra.

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Cool Down:

milkshakes,

revisited

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Reflections

Can I explain the solution to a system of equations in a real-world context?
Can I explain what a system of equations is?
Can I make graphs to find an ordered pair that two real-world situations have in common?

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Practice Problems

Must do: All problems

ALL PERIODS click here to submit homework.

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