Q: Which of the following systems has infinitely many solutions?

$$\begin{cases} y = 2x + 1 \\ y = 2x + 1 \end{cases}$$

Q: A chemist is mixing two solutions, one with a concentration of 10% and another with 30%. If she wants 100 mL of a 20% solution, which system of equations represents the scenario?

$$\begin{cases} x + y = 100 \\ 0.10x + 0.30y = 20 \end{cases}$$

Q: If the solution to a system of equations is $$(4, -2)$$, which of the following systems could it be a solution for?

$$\begin{cases} x + y = 2 \\ 2x - y = 10 \end{cases}$$

Q: A store sells pens for $$\2$ each and notebooks for $$\$$3$$ each. If a customer buys 5 items for $$\12$, which system of equations represents this situation?

$$\begin{cases} x + y = 5 \\ 2x + 3y = 12 \end{cases}$$

Q: Which of the following systems is inconsistent?

$$\begin{cases} y = 2x + 1 \\ y = 2x - 3 \end{cases}$$

Question 1

Which of the following best describes the solution to the system of equations $$\begin{cases} y = 2x + 1 \ y = -x + 4 \end{cases}$$ when solved by graphing?

Accepted Answer

The lines intersect at $$x = 1, y = 3$$ and have one solution.

Question 2

Given the system $$\begin{cases} 2x + 3y = 6 \ x - y = 1 \end{cases}$$, which of the following points is a solution?

Accepted Answer

$$(3, 1)$$

Question 3

If a system of equations is graphed and the lines are parallel, what can you conclude about the system?

Accepted Answer

There is no solution.

Question 4

Which of the following systems has infinitely many solutions?

Accepted Answer

$$\begin{cases} y = 2x + 1 \ y = 2x + 1 \end{cases}$$

Question 5

A chemist is mixing two solutions, one with a concentration of 10% and another with 30%. If she wants 100 mL of a 20% solution, which system of equations represents the scenario?

Accepted Answer

$$\begin{cases} x + y = 100 \ 0.10x + 0.30y = 20 \end{cases}$$

Question 6

Determine whether the point $$(2, 3) is a $$solution to the system $$\begin{cases} 2x + y = 7 \ x - y = -1 \end{cases}.$$

Accepted Answer

Yes, it is a solution to both equations.

Question 7

If the solution to a system of equations is $$(4, -2)$$, which of the following systems could it be a solution for?

Accepted Answer

$$\begin{cases} x + y = 2 \ 2x - y = 10 \end{cases}$$

Question 8

A store sells pens for $$\2$ each and notebooks for $$\$$3$$ each. If a customer buys 5 items for $$\12$, which system of equations represents this situation?

Accepted Answer

$$\begin{cases} x + y = 5 \ 2x + 3y = 12 \end{cases}$$

Question 9

Which of the following is the correct graph for the system $$\begin{cases} y = x + 2 \ y = -x + 2 \end{cases}$$?

Accepted Answer

Two lines intersecting at $$(0, 2)$$

Question 10

If a system of equations has no solution, what does this mean about the graphs of the equations?

Accepted Answer

The lines are parallel and never intersect.

Question 11

Solve the system $$\begin{cases} x + y = 5 \ x - y = 1 \end{cases} by $$graphing. What is the solution?

Accepted Answer

$$(3, 2)$$

Question 12

Which of the following points is NOT a solution to $$y = 2x - 1$$?

Accepted Answer

$$(2, 2)$$

Question 13

A medical researcher is tracking the number of patients in two clinics. Clinic A starts with 20 patients and gains 2 per week. Clinic B starts with 10 patients and gains 4 per week. After how many weeks will the clinics have the same number of patients?

Accepted Answer

$$5$$ weeks

Question 14

Which of the following systems is inconsistent?

Accepted Answer

$$\begin{cases} y = 2x + 1 \ y = 2x - 3 \end{cases}$$

Question 15

If the solution to a system is $$(0, -3)$$, which equation must be true?

Accepted Answer

$$y = -3$$

Solving Systems of Linear Equations by Graphing and Substitution

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