Systems of Two Linear Equations in Two Variables MCQ Quiz - Objective Question with Answer for Systems of Two Linear Equations in Two Variables - Download Free PDF
Last updated on Apr 20, 2025
Latest Systems of Two Linear Equations in Two Variables MCQ Objective Questions
Systems of Two Linear Equations in Two Variables Question 1:
A concert hall sells tickets for
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Systems of Two Linear Equations in Two Variables Question 1 Detailed Solution
Systems of Two Linear Equations in Two Variables Question 2:
What is the x-coordinate of the intersection point of the lines
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Systems of Two Linear Equations in Two Variables Question 2 Detailed Solution
Systems of Two Linear Equations in Two Variables Question 3:
Given the system
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Systems of Two Linear Equations in Two Variables Question 3 Detailed Solution
First, solve the system of equations. From
Systems of Two Linear Equations in Two Variables Question 4:
A chemist mixes two solutions to create a mixture. If
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Systems of Two Linear Equations in Two Variables Question 4 Detailed Solution
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The point
Systems of Two Linear Equations in Two Variables Question 5:
Two friends share the cost of a gift. The total cost is represented by
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Systems of Two Linear Equations in Two Variables Question 5 Detailed Solution
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Top Systems of Two Linear Equations in Two Variables MCQ Objective Questions
x2 + y + 7 = 7
20x + 100 - y = 0
The solution to the given system of equations is (x, y). What is the value of x?
Answer (Detailed Solution Below) -10
Systems of Two Linear Equations in Two Variables Question 6 Detailed Solution
Download Solution PDFy = x2 + 3x - 7
y - 5x + 8 = 0
How many solutions are there to the system of equations above?
A. There are exactly 4 solutions.
B. There are exactly 2 solutions.
C. There is exactly 1 solution.
D. There are no solutions.
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Systems of Two Linear Equations in Two Variables Question 7 Detailed Solution
Download Solution PDFChoice C is correct. The second equation of the system can be rewritten as y = 5x - 8. Substituting 5x - 8 for y in the first equation gives 5x - 8 = x2 + 3x - 7. This equation can be solved as shown below:
x2 + 3x - 7 - 5x + 8 = 0
x2 - 2x + 1 = 0
(x - 1)2 = 0
x = 1
Substituting 1 for x in the equation y = 5x - 8 gives y = -3. Therefore, (1, -3) is the only solution to the system of equations.
Choice A is incorrect. In the xy-plane, a parabola and a line can intersect at no more than two points. Since the graph of the first equation is a parabola and the graph of the second equation is a line, the system cannot have more than 2 solutions. Choice B is incorrect. There is a single ordered pair (x, y) that satisfies both equations of the system. Choice D is incorrect because the ordered pair (1, -3) satisfies both equations of the system.
y = x2 + 2x + 1
x + y + 1 = 0
If (x1, y1) and (x2, y2) are the two solutions to the system of equations above, what is the value of y1 + y2?
A. -3
B. -2
C. -1
D. 1
Answer (Detailed Solution Below)
Systems of Two Linear Equations in Two Variables Question 8 Detailed Solution
Download Solution PDFChoice D is correct. The system of equations can be solved using the substitution method. Solving the second equation for y gives y = -x - 1. Substituting the expression -x - 1 for y into the first equation gives -x - 1 = x2 + 2x + 1. Adding x + 1 to both sides of the equation yields x2 + 3x + 2 = 0. The left-hand side of the equation can be factored by finding two numbers|whose sum is 3 and whose product is 2, which gives (x + 2)(x + 1) = 0. Setting each factor equal to 0 yields x + 2 = 0 and x + 1 = 0, and solving for x yields x = -2 or x = -1. These values of x can be substituted for x in the equation y = -x - 1 to find the corresponding y-values: y = -(-2) - 1 = 2 - 1 = 1 and y = -(-1) - 1 = 1 - 1 = 0. It follows that (-2, 1) and (-1, 0) are the solutions to the given system of equations. Therefore, (x1, y1) = (-2, 1),(x2, y2) = (-1,0), and y1 + y2 = 1 + 0 = 1.
Choice A is incorrect. The solutions to the system of equations are (x1, y1) = (-2, 1) and (x2, y2) = (-1, 0). Therefore, -3 is the sum of the y-coordinates of the solutions, not the sum of the y-coordinates of the solutions. Choices B and C are incorrect and may be the result of computation or substitution errors.
x - y = 1
x + y = x2 - 3
Which ordered pair is a solution to the system of equations above?
A. (1 + √3, √3)
B. (√3, -√3)
C. (1 + √5, √5)
D. (√5, -1 + √5)
Answer (Detailed Solution Below)
Systems of Two Linear Equations in Two Variables Question 9 Detailed Solution
Download Solution PDFChoice A is correct. The solution to the given system of equations can be found by solving the first equation for x, which gives x = y + 1, and substituting that value of x into the second equation which gives
y + 1 + y = (y + 1)2 - 3. Rewriting this equation by adding like terms and expanding (y + 1)2 gives 2y + 1 = y2 + 2y - 2. Subtracting 2y from both sides of this equation gives 1 = y2 - 2. Adding to 2 to both sides of this equation gives 3 = y2. Therefore, it follows that y = ±√3. Substituting √3 for y in the first equation yields x - √3 = 1. Adding √3 to both sides of this equation yields x = 1 + √3. Therefore, the ordered pair (1 + √3, √3) is a solution to the given system of equations.
Choice B is incorrect. Substituting √3 for x and -√3 for y in the first equation yields √3 - (-√3) = 1, or 2√3 = 1, which isn't a true statement. Choice C is incorrect. Substituting 1 + √5 for x and √5 for y in the second equation yields (1 + √5) + √5 = (1 + √5)2 - 3, or 1 + 2√5 = 2√5 + 3, which isn't a true statement. Choice D is incorrect. Substituting √5 for x and (−1 + √5) for y in the second equation yields √5 + (-1 + √5) = (√5)2 - 3, or 2√5 - 1 = 2, which isn't a true statement.
In the xy-plane, a line with equation 2y = 4.5 intersects a parabola at exactly one point. If the parabola has equation y = -4x2 + bx, where b is a positive constant, what is the value of b?
Answer (Detailed Solution Below) 6
Systems of Two Linear Equations in Two Variables Question 10 Detailed Solution
Download Solution PDFy = x2
2y + 6 = 2(x + 3)
If (x, y) is a solution of the system of equations above and x > 0, what is the value of xy?
A. 1
B. 2
C. 3
D. 9
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Systems of Two Linear Equations in Two Variables Question 11 Detailed Solution
Download Solution PDFChoice A is correct. Substituting x2 for y in the second equation gives 2(x2) + 6 = 2(x + 3). This equation can be solved as follows:
2x2 + 6 = 2x + 6 | Apply the distributive property. |
2x2 + 6 - 2x - 6 = 0 | Subtract 2x and 6 from both sides of the equation. |
2x2 - 2x = 0 | Combine like terms. |
2x(x - 1) = 0 | Factor both terms on the left side of the equation by 2x. |
Thus, x = 0 and x = 1 are the solutions to the system. Since x > 0, only x = 1 needs to be considered. The value of y when x = 1 is y = x2 = 12 = 1. Therefore, the value of xy is (1)(1) = 1.
Choices B, C, and D are incorrect and likely result from a computational or conceptual error when solving this system of equations.
x2 = 6x + y
y = -6x + 36
A solution to the given system of equations is (x, y). Which of the following is a possible value of xy?
A. 0
B. 6
C. 12
D. 36
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Systems of Two Linear Equations in Two Variables Question 12 Detailed Solution
Download Solution PDFChoice A is correct. Solutions to the given system of equations are ordered pairs (x,y) that satisfy both equations in the system. Adding the left-hand and right-hand sides of the equations in the system yields x2 + y = 6x + -6x + y + 36, or x2 + y = y + 36. Subtracting y from both sides of this equation yields x2 = 36. Taking the square root of both sides of this equation yields x = 6 and x = -6. Therefore, there are two solutions to this system of equations, one with an x-coordinate of 6 and the other with an x-coordinate of -6. Substituting 6 for x in the second equation yields y = -6(6) + 36, or y = 0; therefore, one solution is (6, 0). Similarly, substituting -6 for x in the second equation yields y = -6(-6) + 36, or y = 72; therefore, the other solution is (-6, 72). It follows then that if (x, y) is a solution to the system, then possible values of xy are (6)(0) = 0 and (-6)(72) = -432. Only 0 is among the given choices.
Choice B is incorrect. This is the x-coordinate of one of the solutions, (6, 0). Choice C is incorrect and may result from conceptual or computational errors. Choice D is incorrect. This is the square of the x-coordinate of one of the solutions, (6, 0).
x + y = 17x
y = 72
If one solution to the system of equations above is (x, y), what is one possible value of x?
Answer (Detailed Solution Below) 8 - 9
Systems of Two Linear Equations in Two Variables Question 13 Detailed Solution
Download Solution PDFIn the xy-plane, what is the y-coordinate of the point of intersection of the graphs of y = (x - 1)2 and y = 2x - 3?
Answer (Detailed Solution Below) 1
Systems of Two Linear Equations in Two Variables Question 14 Detailed Solution
Download Solution PDFy = x2 - 4x + 4
y = 4 - x
If the ordered pair (x, y) satisfies the system of equations above, what is one possible value of x?