1. Jeremiah is making a shake by mixing two different protein powders, measured in ounces. The strawberry-flavored powder has 4 grams of protein per ounce. The banana-flavored powder has 3 grams of protein per ounce. He wants the drink to have a total of 6 ounces of powder and contain 22 grams of protein.
A Create a system of linear equations to represent the situation that will determine the exact number of ounces needed for each powder.
B Based on the system you wrote, which algebraic method of solving systems of equations will you choose to solve the system? Explain your choice.
C Solve the system algebraically, showing your work. Then, interpret the solution by explaining what your solution represents in the context of the problem.
2. A system of equations is given below.
2x + 7y = 1
-3x – 4y = 5
A Create an equivalent system of equations by replacing the first equation by multiplying the first equation by an integer other than 1, and adding it to the second equation.
B Use any method to solve the equivalent system of equations (the new first equation with the original second equation).
C Prove that the solution for the equivalent system is the same as the solution for the original system of equations.
3. Several systems of equations are given below.
y = 6x – 1.5
y = –6x + 1.5
x + 3y = –6
2x + 6y = 3
2x –y = 5
6x – 3y = 15
A Which system of equations is consistent-independent? How many solutions will the system of equations have? Explain your answers.
B Which system of equations is consistent-dependent? How many solutions will the system of equations have? Explain your answers.
C Which system of equations is inconsistent-independent? How many solutions will the system of equations have? Explain your answers.
4. In a biological lab, the cell growth rate of two different organisms is tracked and recorded each week. Given the growth rate, the number of organisms can be determined using the following equations:
s(x) = 100 + 23x
m(x) = 90(1.2x)
A Complete the table of values.
B Use the table to determine at approximately which point the number of cells will be the same for each organism.
C Graph the system of equations, and show the point of intersection.
D Explain what the points graphed for each line represent.
E Explain how you can determine the solution to the equation 100 + 23x = 90(1.2x) using the graph from part c.
F Find the point(s) of intersection, and explain what the intersection represents in the context of the problem.
5. A private school opened in 2005 with an initial enrollment of 85 students. The enrollment has increased by an average of 18 students each year since the school opened. A nearby public school had an enrollment of 95 in 2005, and its enrollment has increased by an average of 15 students per year.
A Let t = time in years, with t = 0 representing the year 2005. Let f(t) = the number of students enrolled at the private school and g(t) = the number of students enrolled at the public school. Create the two functions to represent the situation.
B Graph the system of equations to determine the the point of intersection.
C Interpret the point of intersection in the context of the problem.