SWBAT estimate a solution to a system of equations by graphing. SWBAT solve the system for an exact solution by substitution.

Students discover both the value and the shortcomings of using graphing to solve systems when the intersection point does not have integer coordinates.

20 minutes

Today's Warm Up, Can you save the diver in time?, accesses students' prior knowledge of solving a system of equations by graphing. The problem models a situation where the approximate solution is shown by using a graph, but not the exact solution. This circumstance allows me to introduce solving a system of equations by substitution when an exact solution is needed. Students have been introduced to solving by substitution once in the previous lesson (see the Power Washer Problem).

**Instructional Note**: The Save the Diver Problem was difficult for some students because of the need to scale the graph. So, I selected a student who had the correct answer to share their work on the document camera for all to see.

After sharing the student example of solving the diver problem by graphing, most students realized that we could estimate that the diver would be saved between 6 and 7 minutes, but that the exact time could not be identified by graphing. The solution to the student example shows that the diver is saved between 6 and 7 minutes at a depth between 180 and 200 feet below the surface of the water. The student does not label the y coordinates as negative in the fourth quadrant to represent the sub descending. The student presented as if the numbers were negative, but did not label it correctly.

In the video below, I demonstrate how to write the 2 equations and solve the system of equations using substitution.

20 minutes

I use this T-Mobile vs. AT&T problem to reinforce the objective for this lesson: **to use the substitution method to solve a system of equations.** I want the students to appreciate how substitution helps one find an exact solution. The most difficult part of this problem for students is to write the equations. I walk around the room monitoring the students work, and once the equations are set up, students seem to be able to solve using substitution.

**Modeling the Plans**

- In order to write the equation to represent the plan for T-mobile, students must combine the $20 monthly fee and the $70 a month fee for data. The $20 and the $70 combine to represent the slope or constant rate of change of $90 per month. The initial fee (the y-intercept) is $99.
- The AT&T plan combines the fee of the phone in the monthly fee, and has no set up fee. Therefore it's equation has a slope of $110, and a y intercept of 0.
- I normally allow the students to choose the method to write the equation, but in this lesson I emphasize to students that increasing and decreasing situations are easily written in slope intercept form. Increasing and decreasing situations have an initial point (the y-intercept) and an increasing or decreasing constant rate of change (the slope).

**Analyzing the Plans**

- The equation for the AT&T plan is y=90x+99
- The equation for the T-mobile plan is y=110x
- The AT&T plan is cheaper under 4.95 months
- The plans both cost the same at exactly 4.95 months (approximately 5 months)
- After the initial fee of $99 is paid, the $90 a month for T-mobile is cheaper over time than the $110 a month for AT&T.
- T-mobile is cheaper at 24 months, totaling $2259
- AT&T is more expensive at 24 months, totaling $2640

Solving this system by graphing provides a good visual and an approximate answer. However, the Substitution method gives the exact (x,y) solution of (4.95, 544.50). Since the phone bill is paid monthly,the whole numbers are the relevant numbers. The T-mobile plan will be cheaper at four months, and AT&T is cheaper at 5 months.

10 minutes

I use this Exit ticket as a formative assessment to check for student understanding of substitution. This problem also introduces students to a different type of problem using profit, revenue, and break-even point. Students did struggle with the 2 equations being set to E and R to represent expense and revenue. The variable x represents the number of stereos.

The break even point for the company is at exactly 18.75 stereos (which is not possible), totaling a revenue and expense of $3375. The company has a revenue of $3420, and a profit of $45 at 19 stereos. The company spends more on expenses prior to18.75 stereos or approximately 19 stereos than the revenue that it earns.