A system of equations

<object:standard:macc.912.a–rei.3.5>A system of equations is shown below:

x + 3y = 5      (equation 1)

7x – 8y = 6     (equation 2)

A student wants to prove that if equation 2 is kept unchanged and equation 1 is replaced with the sum of equation 1 and a multiple of equation 2, the solution to the new system of equations is the same as the solution to the original system of equations. If equation 2 is multiplied by 1, which of the following steps should the student use for the proof?

 Show that the solution to the system of equations 3x + y = 5 and 8x –7y = 6 is the same as the solution to the given system of equations Show that the solution to the system of equations 8x – 5y = 11 and 7x – 8y = 6 is the same as the solution to the given system of equations Show that the solution to the system of equations 15x + 13y = 17 and 7x – 8y = 6 is the same as the solution to the given system of equations Show that the solution to the system of equations –13x + 15y = 17 and 7x – 8y = 6 is the same as the solution to the given system of equations

Question 2 

(05.02)<object:standard:macc.912.a–rei.3.6>

Solve the following system of equations:

3x - 2y = 6

6x - 4y = 12

 (0, 0) (6, 12) Infinitely many solutions No solutions

Question 3 

(05.01)<object:standard:macc.912.a–rei.3.6>

Which graph below shows a system of equations with infinitely many solutions?

Question 4 

(05.02)<object:standard:macc.912.a–rei.3.6>

A system of equations is shown below:

x + y = 3

2x – y = 6

The x-coordinate of the solution to this system of equations is _____.

Numerical Answers Expected!

Answer for Blank 1:

Question 5 

(05.01)<object:standard:macc.912.a–rei.3.6>

The two lines, X and Y, are graphed below:

Determine the solution and the reasoning that justifies the solution to the systems of equations.

 (2, 7), because this point is true for both the equations (4, –6), because this point lies only on one of the two lines (4, –6), because this point makes both the equations true (2, 7), because the lines intersect the x-axis at these points

Question 6 

(05.06)<object:standard:macc.912.a–ced.1.3>

Nick works two jobs to pay for college. He tutors for $15 per hour and also works as a bag boy for $8 per hour. Due to his class and study schedule, Nick is only able to work up to 20 hours per week but must earn at least $150 per week. If t represents the number of hours Nick tutors and b represents the number of hours he works as a bag boy, which system of inequalities represents this scenario?

 t + b  20

15t + 8b = 150 t + b  20

15t + 8b  150 t + b  20

15t + 8b  150 None of the systems shown represent this scenario.

Question 7 

(05.02)<object:standard:macc.912.a–rei.3.5>

Two systems of equations are shown below:

System ASystem B2x + y = 5-10x + 19y = -1-4x + 6y = -2-4x + 6y = -2

Which of the following statements is correct about the two systems of equations?

 They will have the same solutions because the first equation of System B is obtained by adding the first equation of System A to 2 times the second equation of System A. They will have the same solution because the first equation of System B is obtained by adding the first equation of System A to 3 times the second equation of System A. The value of x for System B will be –5 times the value of x for System A because the coefficient of x in the first equation of System B is –5 times the coefficient of x in the first equation of System A. The value of x for System A will be equal to the value of y for System B because the first equation of System B is obtained by adding –12 to the first equation of System A and the second equations are identical.

Question 8 

(05.06)<object:standard:macc.912.a–rei.4.12>

A system of linear inequalities is shown below:

y – x > 0

x + 1 < 0

Which of the following graphs best represents the solution set to this system of linear inequalities?

Question 9 

(05.02)<object:standard:macc.912.a–rei.3.6>

The total price of a shirt and a cap is $11. If the price of the shirt was doubled and the price of the cap was 3 times its original price, the total price of a shirt and a cap would be $25. What is the price of a shirt and a cap?

 The price of a shirt is $9, and the price of a cap is $2. The price of a shirt is $10, and the price of a cap is $1. The price of a shirt is $8, and the price of a cap is $3. The price of a shirt is $7, and the price of a cap is $4.

Question 10 

(05.05)<object:standard:macc.912.a–rei.4.12>

Select the inequality that corresponds to the given graph. 

 4x – 3y > – 12 –x + 4y > 4 4x – 2y < – 8 2x + 4y ≥ – 16

Question 11 

(05.03)<object:standard:macc.912.a–rei.4.10>

The graph of an equation is shown below:

Based on the graph, which of the following represents a solution to the equation?

 (–2,–3) (3, 1) (1, 3) (3, 2)

Question 12 

(05.03)<object:standard:macc.912.a–rei.4.11>

The tables below show the values of f(x) and g(x) for different values of x:

f(x) = 2(3)x 

xf(x)-20.22-10.670216218

g(x) = 3x + 9

xg(x)-29.11-19.33010112218

Based on the tables, what is the solution to the equation 2(3)x = 3x + 9?

 x = 0 x = 2 x = 12 x = 18

Question 13 

(05.03)<object:standard:macc.912.a–rei.4.11>

An equation is shown below:

What is the solution to the equation?

 x = –2 x = –1 x = 1 x = 2

Question 14 

(05.02)<object:standard:macc.912.a–ced.1.3>

Maya had $27. She spent all the money on buying 3 burgers for $x each and 2 sandwiches for $y each. If Maya had bought 2 burgers and 1 sandwich, she would have been left with $11. 

A student concluded that the price of each burger is $5 and the price of each sandwich is $6. Which statement best justifies whether the student's conclusion is correct or incorrect?

 The student's conclusion is correct because the solution to the system of equations 3x + 2y = 11 and 2x + y = 16 is (5, 6). The student's conclusion is incorrect because the solution to the system of equations 3x – 2y = 11 and 2x – y = 16 is (5, 6). The student's conclusion is correct because the solution to the system of equations 3x + 2y = 27 and 2x + y = 16 is (5, 6). The student's conclusion is incorrect because the solution to the system of equations 2x + 3y = 27 and x + 2y = 16 is (5, 6).

Question 15 

(05.06)<object:standard:macc.912.a–rei.4.12>

Look at the graph below:

Which part of the graph best represents the solution set to the system of inequalities y ≥ x + 1 and y + x ≤ –1?

 Part A Part B Part C Part D

Question 16 

(05.05)<object:standard:macc.912.a–rei.4.12>

A graph is shown below:

Which of the following inequalities is best represented by this graph?

 5x + y ≤ 2 5x + y ≥ 2 5x – y ≤ 2 5x – y ≥ 2

Question 17 

(05.01)<object:standard:macc.912.a–rei.3.6>

The graph plots four equations, A, B, C, and D:

Which pair of equations has (0, 8) as its solution?

 Equation A and Equation C Equation B and Equation C Equation C and Equation D Equation B and Equation D

Question 18 

(05.03)<object:standard:macc.912.a–rei.4.11>

The graph below shows two functions:

Based on the graph, what are the approximate solutions to the equation –2x + 8 = (0.25)x?

 8 and 4 1 and 4 –1.7 and 4 1.7 and –4

Question 19 

(05.05)<object:standard:macc.912.a–rei.4.12>

Greg made 4 chairs and 3 tables. Greg only has enough plywood to make at most 15 chairs or tables total. Let x represent the number of more chairs and y represent the number of more tables that Greg can make. Which of the following graphs best represents the relationship between x and y?

Question 20 

(05.02)<object:standard:macc.912.a–rei.3.6>

For the following system, if you isolated x in the first equation to use the Substitution Method, what expression would you substitute into the second equation?

-x - 2y = -4

3x + y = 12

 -2y - 4 2y - 4 2y + 4 -2y + 4

Question 21

(05.05)<object:standard:macc.912.a–rei.4.12>

Sally has only nickels and dimes in her money box. She knows that she has less than $20 in the box. Let x represent the number of nickels in the box and y represent the number of dimes in the box. Which of the following statements best describes the steps to graph the solution to the inequality in x and y?

 Draw a dashed line to represent the graph of 5x + 10y = 2000, and shade the portion above the line for positive values of x and y. Draw a dashed line to represent the graph of 10x + 5y = 2000, and shade the portion above the line for positive values of x and y. Draw a dashed line to represent the graph of 5x + 10y = 2000, and shade the portion below the line for positive values of x and y. Draw a dashed line to represent the graph of 10x – 5y = 2000, and shade the portion below the line for positive values of x and y.

Question 22 

(05.03)<object:standard:macc.912.a–rei.4.10>

Which of the following statements best describes the graph of –5x + 2y = 1?

 It is a curve joining the points (–5, 2), (2, 3), and (4, 1). It is a curve joining the points (–1, –3), (–1, –3), and (1, 5). It is a straight line joining the points (1, 3), (3, 8), and (–3, –7). It is a straight line joining the points (4, –3), (–1, 2), and (–4, 5).

Question 23 

(05.01)<object:standard:macc.912.a–rei.3.6>

A pair of linear equations is shown below:

y = –x + 1

y = 2x + 4

Which of the following statements best explains the steps to solve the pair of equations graphically?

 On a graph, plot the line y = –x + 1, which has y-intercept = –1 and slope = 1, and y = 2x + 4, which has y-intercept = 2 and slope = 4, and write the coordinates of the point of intersection of the two lines as the solution. On a graph, plot the line y = –x + 1, which has y-intercept = 1 and slope = 1, and y = 2x + 4, which has y-intercept = 1 and slope = 4, and write the coordinates of the point of intersection of the two lines as the solution. On a graph, plot the line y = –x + 1, which has y-intercept = 1 and slope = –1, and y = 2x + 4, which has y-intercept = –2 and slope = 2, and write the coordinates of the point of intersection of the two lines as the solution. On a graph, plot the line y = –x + 1, which has y-intercept = 1 and slope = –1, and y = 2x + 4, which has y-intercept = 4 and slope = 2, and write the coordinates of the point of intersection of the two lines as the solution.