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

$$\begin{cases} 12x - 11y = 15 \\ 6x - 5.5y = -13 \end{cases}$$

Question 1

Which of the following ordered pairs is a solution to the system of equations below?

$$\begin{cases} 4x + 5y = 2 \ x - 2y = -6 \end{cases}$$

Accepted Answer

$$(-2,-2)$$

Question 2

Consider the system of equations:

$$\begin{cases} y = 5x + 15 \ y = -7x - 33 \end{cases}$$

What is the solution to this system?

Accepted Answer

$$x = -4,\ y = -5$$

Question 3

Given the system:

$$\begin{cases} 2x + 13y = 6 \ x + 6.5y = 3 \end{cases}$$

Which of the following best describes the solution and its graphical meaning?

Accepted Answer

A unique solution; the two lines intersect at a single point.

Question 4

Solve the following system by substitution:

$$\begin{cases} x = 6y + 9 \ 5x + 7y = -29 \end{cases}$$

What is the solution?

Accepted Answer

$$x = -3,\ y = -2$$

Question 5

Solve the following system by elimination:

$$\begin{cases} 4x - y = 31 \ x + 5y = 13 \end{cases}$$

What is the solution?

Accepted Answer

$$x = 8,\ y = 1$$

Question 6

A store's demand and supply for a product are given by the equations:

$$\begin{cases} p + 4q = 346 \ p - 3q = 178 \end{cases}$$

What is the equilibrium price $$p at $$which the quantity demanded equals the quantity supplied?

Accepted Answer

$$p = 250$$

Question 7

Ms. Walsh invested $$19,000 in two accounts, one yielding 4$$\%$$ interest and the other 8$$\%$$. If she received a total of 920 in $$interest at the end of the year, how much did she invest at $$8\%$$?

Accepted Answer

$$4,000$$

Question 8

A mixture of soybean meal ($$18\%$$ protein) and cornmeal ($$9\%$$ protein) is to be mixed to obtain $$360 lb of a $$mixture that is $$10\%$$ protein. How many pounds of cornmeal should be in the mixture?

Accepted Answer

$$320 lb$$

Question 9

Sarah blends $$110$$ pounds of coffee beans to sell at $$\5.36$ per pound by mixing high-quality beans at $$\$$6.25$$ per pound and cheaper beans at $$\3.00$ per pound. How many pounds of high-quality beans should she use (rounded to the nearest pound)?

Accepted Answer

$$80 lb$$

Question 10

Solve the following system by elimination:

$$\begin{cases} 7x = 21 - 14y \ 6x + 2y = 6 \end{cases}$$

What is the solution?

Accepted Answer

$$x = 3,\ y = 5$$

Question 11

Which of the following systems has infinitely many solutions?

Accepted Answer

$$\begin{cases} 12x - 11y = 15 \ 6x - 5.5y = -13 \end{cases}$$

Question 12

Solve the following system by any convenient method:

$$\begin{cases} 2x = 6y - 10 \ 4x = 14 - 5y \end{cases}$$

What is the solution?

Accepted Answer

$$x = 1,\ y = 2$$

Question 13

A business finds that the break-even point for a product occurs when supply equals demand. If the supply is $$p + 4q = 346$$ and the demand is $$p - 3q = 178$$, what is the equilibrium quantity $$q$$?

Accepted Answer

$$q = 24$$

Question 14

Which of the following best describes the graphical meaning of a system of two linear equations that has no solution?

Accepted Answer

The two lines are parallel.

Question 15

Solve the following system by elimination:

$$\begin{cases} 0.2x + 0.5y = 3.5 \ 3x - 2y = 5 \end{cases}$$

What is the solution?

Accepted Answer

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

Question 16

A mixture problem: You want to mix $$18\%$$ protein soybean meal and $$9\%$$ protein cornmeal to get $$360 lb of a$$ $$10\%$$ protein mixture. How many pounds of soybean meal should be used?

Accepted Answer

$$40 lb$$

Question 17

Solve the following system by any convenient method:

$$\begin{cases} 3x + 2y = 7 \ 9x + 6y = 0 \end{cases}$$

What is the solution?

Accepted Answer

There is no solution.

Question 18

A system of equations is solved graphically and the two lines coincide. What does this mean about the solutions?

Accepted Answer

There are infinitely many solutions.

Question 19

A real-world application: A company wants to blend $$110$$ pounds of coffee beans to sell at $$\5.36$ per pound by mixing high-quality beans at $$\$$6.25$$ per pound and cheaper beans at $$\3.00$ per pound. How many pounds of cheaper beans should be used (rounded to the nearest pound)?

Accepted Answer

$$30 lb$$

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