Constraints System of Linear Equations

A system of linear equations is given. x − y = 5 x + y = −5 Which statement is TRUE about the solution of the system of linear equations?

• The solution is (0, −5) because this ordered pair represents the y-intercepts for both equations.

• The solution is (−5, 0) because this ordered pair represents the x-intercepts for both equations.

• The solution is (0, −5) because this ordered pair represents the point of intersection for both lines in the system of linear equations.

• The solution is (−5, 0) because this ordered pair represents the point of intersection for both lines in the system of linear equations.

Jontez and KD sold 100 tickets for a fundraiser. KD sold 24 more tickets than Jontez. The equations that represent the scenario are x + y = 100 and x − y = 24. Which statement is TRUE about the number of tickets sold?

• There is 1 solution to the system of equations, which shows that KD sold 62 tickets.

• There is 1 solution to the system of equations, which shows that KD sold 76 tickets.

• There are infinitely many solutions to the system of equations because there are infinitely many combinations of tickets sold that have a sum of 100.

• There are infinitely many solutions to the system of equations because there are infinitely many combinations of tickets sold that have a difference of 24.

The system of linear equations represents the total sales an airline wants to have for 2 different flights represented by the number of economy seats sold, x, and first class seats sold, y. 400x + 600y = 8,000 400x + 600y = 16,000 What statement BEST explains the number of solutions for the system of linear equations?

• The system of linear equations has 1 solution because 400x + 600y can equal 2 different numbers.

• The system of linear equations has no solutions because 400x + 600y cannot equal 2 different numbers.

• The system of linear equations has 1 solution because the second equation is a multiple of the first equation.

• The system of linear equations has no solutions because the second equation is a multiple of the first equation.

Brittany, Keondra, and Aniyah are heading along the same straight sidewalk towards a park gate. The graph shows Brittany's distance, y, in feet from the gate after x minutes. The equation y = −300x + 700 represents Keondra's distance, y, in feet from the gate after x minutes. The equation y = −300x + 600 represents Aniyah's distance, y, in feet from the gate after x minutes. Which statement is TRUE about when Brittany, Keondra, and Aniyah are predicted to pass by each other?

• Keondra and Brittany will pass by each other before Aniyah and Brittany will pass by each

  other, but Aniyah and Keondra will not pass by each other.

• Aniyah and Brittany will pass by each other before Keondra and Brittany will pass by each

  other, but Aniyah and Keondra will not pass by each other.

• Keondra and Brittany will pass by each other, then Aniyah and Brittany will pass by each other,

  and then Aniyah and Keondra will pass by each other.

• Aniyah and Brittany will pass by each other, then Keondra and Brittany will pass by each other,

  and then Aniyah and Keondra will pass by each other.

Which THREE systems have a solution with an x-value that is less than the y-value?

• 6x -8y = 6 and

  y = 2x + 8

• 
• -3x + 5y = -5 and

  x - 7y = -25

• 
• 6x - 8y = 6 and

  y = 2x + 8

Which TWO systems of equations have exactly one solution?

• 6x – 3y = 3 2x – y = 4

• y = –2x – 8 y = x + 1

• 6x – 3y = 12 2x – y = 4

• x – 4y = 8 3x + 2y = 10

• y = 2x + 1 y = 2x – 2

Two students solved a system of equations. Original system: 5x - 11y = -34 8x - 12y = -4 Student 1: 8x = 12y - 4 x = 3/2 y - 1/2 5(3/2 y - 1/2) - 11y = -34 15/2 y - 5/2 - 11y = -34 -7/2 y = -63/2 y = 9 Student 2: -40x + 88y = -42 40x - 60y = -20 28y = -62 y = -2 3/14 Why did the students come up with different values for the variable y?

• Student 1 incorrectly transformed the first equation.

• Student 1 incorrectly divided in the last step.

• Student 2 incorrectly transformed the first equation.

• Student 2 incorrectly divided in the last step.

What is a
feasible region?

• A region when the graph of a system of constraints is a polygonal region. 

• The intersection of the graphs in a system of constraints.

• A system of inequalities that that forms a region that is open.

• A system of equations that has an infinite number of solutions.

What is
Cramer's Rule?

• A square matrix that, when multiplied by another matrix, equals that same matrix. If A is any n X n matrix and I is the n X n identity matrix, then A ⋅ I = A and I ⋅ A  =  A.

•  A matrix that contains only the coefficients of a system of equations.

•  A method that uses determinants to solve a system of linear equations.

• A method of solving a system of equations in which one equation is solved for one variable in terms of the other.

What is a
Coefficient Matrix?

• A matrix that contains only the coefficients of a system of equations.

• A square matrix that, when multiplied by another matrix, equals that same matrix. If A is any n X n matrix and I is the n X n identity matrix, then A ⋅ I = A and I ⋅ A  =  A.

• Two n X n matrices whose product is the identity matrix.

• A matrix that contains the constants of a system.