Q: Which of the following systems has no solution?

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

Q: Which of the following systems is best solved by the substitution method?

$$\begin{cases} y = 3x + 2 \\ 2x + y = 8 \end{cases}$$

Q: If the solution to a system is $$(x, y) = (5, -2)$$, which of the following systems could it be?

$$\begin{cases} x + y = 3 \\ x - y = 7 \end{cases}$$

Question 1

Which method is used to solve the system of equations below?

$$\begin{cases} 2x + y = 7 \ x - y = 1 \end{cases}$$

Accepted Answer

Elimination

Question 2

Solve the following system using the elimination method:

$$\begin{cases} x + y = 6 \ x - y = 2 \end{cases}$$

Accepted Answer

$$x = 4,\ y = 2$$

Question 3

Given the system $$\begin{cases} 3x + 2y = 12 \ x - 2y = 0 \end{cases}$$, what is the solution?

Accepted Answer

$$(2, 3)$$

Question 4

If you graph the equations $$y = 2x + 1$$ and $$y = -x + 4$$, what is the solution to the system?

Accepted Answer

$$(1, 3)$$

Question 5

A chemist mixes two solutions, one with a concentration of $$x$$ and another with $$y. If$$ $$x + y = 10$$ and $$2x - y = 4$$, what are the values of $$x$$ and $$y$$?

Accepted Answer

$$x = 7,\ y = 3$$

Question 6

Which of the following systems has no solution?

Accepted Answer

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

Question 7

A store sells pens and pencils. Pens cost $$x$$ dollars and pencils cost $$y$$ dollars. If 3 pens and 2 pencils cost $$11$$, and 2 pens and 4 pencils cost $$12$$, what are the costs of a pen and a pencil?

Accepted Answer

$$x = 2,\ y = 2.5$$

Question 8

Which graph represents the solution to the system $$y = x + 1$$ and $$y = -x + 5$$?

Accepted Answer

The lines intersect at $$(2, 3)$$

Question 9

If a system of equations has infinitely many solutions, what does this mean about the two equations?

Accepted Answer

They are the same line

Question 10

Solve the following system using substitution:

$$\begin{cases} y = 2x + 3 \ x + y = 9 \end{cases}$$

Accepted Answer

$$x = 2,\ y = 7$$

Question 11

A medical researcher is mixing two solutions for a test. Solution A contains $$x mg of a $$drug, and Solution B contains $$y mg. If$$ $$x + y = 15$$ and $$x - y = 5$$, what are the values of $$x$$ and $$y$$?

Accepted Answer

$$x = 10,\ y = 5$$

Question 12

Which of the following systems is best solved by the substitution method?

Accepted Answer

$$\begin{cases} y = 3x + 2 \ 2x + y = 8 \end{cases}$$

Question 13

A system of equations is given by $$\begin{cases} 2x + 3y = 12 \ 4x + 6y = 24 \end{cases}. $$What is true about the solution?

Accepted Answer

There are infinitely many solutions

Question 14

If the solution to a system is $$(x, y) = (5, -2)$$, which of the following systems could it be?

Accepted Answer

$$\begin{cases} x + y = 3 \ x - y = 7 \end{cases}$$

Question 15

A business has two types of expenses, fixed ($$x) $$and variable ($$y). If$$ $$x + y = 1000$$ and $$x - y = 200$$, what are the values of $$x$$ and $$y$$?

Accepted Answer

$$x = 600,\ y = 400$$

Solving Systems of Equations – Algebra Study Guidance

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