Q: Which of the following systems has no solution?

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

Q: Which of the following systems represents a dependent system?

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

Question 1

Which of the following best describes a consistent system of two linear equations in two variables?

Accepted Answer

A system with at least one solution; the lines intersect at one or infinitely many points.

Question 2

Given the system: 
$$\begin{cases} 2x - 3y = 4 \ 2x + y = 4 \end{cases}
Is $$the ordered pair $$(1, 2) a $$solution to the system?

Accepted Answer

No, because it does not satisfy either equation.

Question 3

Solve the following system by substitution:
$$\begin{cases} 3x + 2y = 4 \ 2x + y = 1 \end{cases}$$

Accepted Answer

$$(2, -1)$$

Question 4

A company manufactures running shoes with a fixed cost of $$300,000 and a variable cost of 30$ per pair. Each pair sells for 80$. What is the break-even point (number of pairs sold)?

Accepted Answer

$$6,000$$

Question 5

Which of the following systems has no solution?

Accepted Answer

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

Question 6

Solve the following system by the addition method:
$$\begin{cases} 4x + 5y = 3 \ 2x - 3y = 7 \end{cases}$$

Accepted Answer

Question 7

Which of the following ordered triples is a solution to the system:
$$\begin{cases} x - 2y + 3z = 22 \ 2x - y - z = 5 \ 3x - 5y + 5z = 32 \end{cases}$$?

Accepted Answer

$$(-1, -4, 5)$$

Question 8

Solve the nonlinear system:
$$\begin{cases} x^2$ + y^2$ = 49 \ y^2$ - 6x = 49 \end{cases}$$

Accepted Answer

$$(7, 0)$$ and $$(0, -7)$$

Question 9

Which of the following best describes the graph of the solution set for the inequality $$4x - 2y \geq 8$$?

Accepted Answer

A solid line with shading to the right of the line.

Question 10

Let $$x$$ and $$y be $$two numbers such that $$x + y = 16$$ and $$xy = 63. $$What are the numbers?

Accepted Answer

$$7$$ and $$9$$

Question 11

Which of the following is the correct first step to solve the system by substitution?
$$\begin{cases} 2x + y = 1 \ 3x + 2y = 4 \end{cases}$$

Accepted Answer

Solve the first equation for $$y in $$terms of $$x.$$

Question 12

Which of the following systems represents a dependent system?

Accepted Answer

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

Question 13

A system of inequalities is given by:
$$\begin{cases} x - 3y \u003c 6 \ 2x + 3y \geq -6 \end{cases}$$
Which region represents the solution set?

Accepted Answer

The region where the shaded areas of both inequalities overlap.

Question 14

Solve the following system of three equations:
$$\begin{cases} x + y = -4 \ y - z = 1 \ 2x + y + 3z = -21 \end{cases}$$

Accepted Answer

$$(-3, -1, -2)$$

Question 15

Which of the following best describes the solution set for the system:
$$\begin{cases} x^2$ + y^2$ \u003c 4 \ y - x^2$ \geq 0 \end{cases}$$

Accepted Answer

The region inside the circle and above the parabola.

Question 16

Which of the following is NOT a step in solving a system of linear equations in three variables by elimination?

Accepted Answer

Graph the three equations on a number line.

Question 17

Given the system:
$$\begin{cases} x + y = 1 \ (x-1)^2$ + (y+2)^2$ = 10 \end{cases}$$
What are the solutions?

Accepted Answer

$$(2, -1)$$ and $$(-1, 2)$$

Question 18

Which of the following best describes the graph of $$x \u003c -3$$?

Accepted Answer

A dashed vertical line at $$x = -3$$ with shading to the left.

Systems of Equations and Inequalities: Study Notes for Precalculus

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