Q: Which of the following systems is inconsistent?

$$\begin{cases} x + y = 2 \\ x + y = 4 \end{cases}$$

Question 1

Which of the following best describes the solution to a system of two equations in two variables?

Accepted Answer

A pair of real numbers that satisfies both equations.

Question 2

Given the system of equations: 
$$\begin{cases} 2x - 3y = -23 \ x + y = 0 \end{cases}$$
What is the solution using the substitution method?

Accepted Answer

$$(x, y) = \left( -\frac{23}{5}, \frac{23}{5} \right)$$

Question 3

Which method is most efficient for solving a system where one equation is already solved for a variable?

Accepted Answer

Substitution method

Question 4

A business uses the following system to model costs and revenue: 
$$C = 5x + 2y ($$cost), $$R = 8x + 4y ($$revenue). If the break-even point is when $$C = R$$, what is the relationship between $$x$$ and $$y at $$break-even?

Accepted Answer

$$3x + 2y = 0$$

Question 5

Solve the following system using the elimination method:
$$\begin{cases} 3x - 2y = 8 \ 5x + 4y = 28 \end{cases}$$

Accepted Answer

$$(x, y) = (4, 2)$$

Question 6

If a system of equations results in $$0 = 0$$ after elimination, what does this indicate about the system?

Accepted Answer

There are infinitely many solutions.

Question 7

Which of the following systems is inconsistent?

Accepted Answer

$$\begin{cases} x + y = 2 \ x + y = 4 \end{cases}$$

Question 8

A chemist is mixing two solutions. The system $$\begin{cases} x + y = 10 \ 2x + 3y = 24 \end{cases}$$ models the total volume and concentration. How much of each solution should be used?

Accepted Answer

$$x = 6, y = 4$$

Question 9

Given the system $$\begin{cases} y = 3 - 7x \ y = x^2$ \end{cases}$$, how many real solutions does the system have?

Accepted Answer

Two

Question 10

Which of the following is the correct first step in solving $$\begin{cases} 2x - 3y = 5 \ -6x + 9y = -15 \end{cases}$$ using elimination?

Accepted Answer

Multiply the first equation by 3.

Question 11

A system of equations models the intersection of a line and a parabola. What does it mean if the system has two solutions?

Accepted Answer

The line intersects the parabola at two points.

Question 12

Solve for $$x in $$the system $$\begin{cases} y = 3 - 7x \ y = x^2$ \end{cases}.$$

Accepted Answer

$$x = 0.333$$ and $$x = 1.5$$

Question 13

Which of the following best describes the graphical solution to a system with infinitely many solutions?

Accepted Answer

The graphs are the same line.

Question 14

If a system of equations in three variables has no solution, what does this mean geometrically?

Accepted Answer

The three planes do not all intersect at a common point.

Question 15

Which of the following is NOT a valid method for solving systems of equations?

Accepted Answer

Integration

Question 16

A medical researcher uses the system $$\begin{cases} x + 2y = 10 \ 3x - y = 5 \end{cases} to $$model dosages. What is the value of $$y$$?

Accepted Answer

$$y = 2$$

Solving Systems of Equations: Substitution, Elimination, and Graphical Methods

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