Systems of Equations in Two Variables MCQ Quiz - Objective Question with Answer for Systems of Equations in Two Variables - Download Free PDF

Last updated on Apr 10, 2025

Latest Systems of Equations in Two Variables MCQ Objective Questions

Systems of Equations in Two Variables Question 1:

A rectangle has a length that is twice its width. If the area of the rectangle is square units, what is the width of the rectangle?

  1. 5
  2. 5
  3. 10
  4. 4

Answer (Detailed Solution Below)

Option 1 : 5

Systems of Equations in Two Variables Question 1 Detailed Solution

Let be the width of the rectangle. The length is . The area is given by . This simplifies to . Dividing by , we have . Taking the square root of both sides gives or . Since width cannot be negative, .

Systems of Equations in Two Variables Question 2:

A recipe calls for the equation . If represents the number of cups of flour and represents the number of cups of sugar, express in terms of .

  1. y =
  2. y =
  3. y =
  4. y =

Answer (Detailed Solution Below)

Option 1 : y =

Systems of Equations in Two Variables Question 2 Detailed Solution

To express in terms of , start with . Subtract from both sides to get . Divide every term by to isolate : . Thus, Option 1 is correct. Option 2 reverses subtraction, Option 3 does not account for division, and Option 4 misconstrues the equation structure. Following the algebraic manipulations accurately results in the correct expression.

Systems of Equations in Two Variables Question 3:

A car rental company charges a flat fee of plus per mile driven. If the total cost is , how many miles were driven?

  1. 320 miles
  2. 3200 miles
  3. 200 miles
  4. 400 miles

Answer (Detailed Solution Below)

Option 1 : 320 miles

Systems of Equations in Two Variables Question 3 Detailed Solution

The cost equation given by the rental company is , where is the number of miles driven. To find , first subtract from both sides: . Then, divide both sides by to solve for : . Therefore, the number of miles driven is 320, making option 1 the correct answer. Other options do not satisfy the equation when plugged in.

Systems of Equations in Two Variables Question 4:

Solve for in the equation .

  1. x = 10 or x = -2
  2. x = 8 or x = -4
  3. x = 12 or x = -12
  4. x = 6 or x = -6

Answer (Detailed Solution Below)

Option 1 : x = 10 or x = -2

Systems of Equations in Two Variables Question 4 Detailed Solution

To solve , we must consider both the positive and negative cases of the absolute value expression.

First, solve the positive case: .

Add 8 to both sides: .

Divide by 2: .

Now, solve the negative case: .

Add 8 to both sides: .

Divide by 2: .

The solutions to the equation are and . Thus, option 1 is correct.

Systems of Equations in Two Variables Question 5:

A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is 70 square meters, what is the width of the garden?

  1. 5
  2. 7
  3. 8
  4. 6

Answer (Detailed Solution Below)

Option 2 : 7

Systems of Equations in Two Variables Question 5 Detailed Solution

Let the width of the garden be meters. Then the length is meters. The area of the rectangle is given by . Expanding this, we have . Rearranging gives , a quadratic equation in standard form. To solve for , we use the quadratic formula , where , , and . The discriminant is . The solutions are . This gives and . The width must be positive, so meters. Therefore, the correct answer is option 2. Options 1, 3, and 4 are not correct because they do not satisfy the area equation.

Top Systems of Equations in Two Variables MCQ Objective Questions

Systems of Equations in Two Variables Question 6:

A farmer has two types of crops. The equation represents the number of acres for each crop. If the farmer plants twice as many acres of crop as crop , how many acres of crop does he plant?

  1. 6
  2. 10
  3. 12
  4. 8

Answer (Detailed Solution Below)

Option 1 : 6

Systems of Equations in Two Variables Question 6 Detailed Solution

We know and . Substituting into the first equation gives . Simplifying yields , which simplifies to . Solving for , we find , with , , and recalculating , which is correct. Therefore, the farmer plants 6 acres of crop .

Systems of Equations in Two Variables Question 7:

If , express in terms of and .

Answer (Detailed Solution Below)

Option 1 :

Systems of Equations in Two Variables Question 7 Detailed Solution

To express in terms of and , begin with the given equation: . By multiplying both sides by , we get . Dividing both sides by gives . Thus, option 1 is correct. Option 2 is incorrect as it implies , which does not match the original equation. Option 3 is incorrect since would imply a multiplication, not a division. Option 4 is incorrect as it suggests is , which doesn't represent the original expression.

Systems of Equations in Two Variables Question 8:

Determine the value of if .

  1. 3, -3
  2. 3, -2.6
  3. -3, 3
  4. 2.6, -3

Answer (Detailed Solution Below)

Option 2 : 3, -2.6

Systems of Equations in Two Variables Question 8 Detailed Solution

For the absolute value equation , we consider the two cases for the absolute value:

1.

2.

For the first equation, :

Add to both sides:

Divide by :

For the second equation, :

Add to both sides:

Divide by :

Thus, the solutions are and , which correspond to option 2.

Systems of Equations in Two Variables Question 9:

Find the value of such that the system of equations and has exactly one solution.

  1. 5
  2. 8
  3. 3
  4. 6.25

Answer (Detailed Solution Below)

Option 4 : 6.25

Systems of Equations in Two Variables Question 9 Detailed Solution

To find the value of where the line intersects the parabola at one point, equate the equations: . Rearrange to form a quadratic: . For the quadratic to have exactly one solution, the discriminant must be zero. The discriminant is . Simplifying gives , solving for gives . Therefore, the value of is .

Systems of Equations in Two Variables Question 10:

A rectangle has a length that is twice its width. If the area of the rectangle is square units, what is the width of the rectangle?

  1. 5
  2. 5
  3. 10
  4. 4

Answer (Detailed Solution Below)

Option 1 : 5

Systems of Equations in Two Variables Question 10 Detailed Solution

Let be the width of the rectangle. The length is . The area is given by . This simplifies to . Dividing by , we have . Taking the square root of both sides gives or . Since width cannot be negative, .

Systems of Equations in Two Variables Question 11:

A recipe calls for the equation . If represents the number of cups of flour and represents the number of cups of sugar, express in terms of .

  1. y =
  2. y =
  3. y =
  4. y =

Answer (Detailed Solution Below)

Option 1 : y =

Systems of Equations in Two Variables Question 11 Detailed Solution

To express in terms of , start with . Subtract from both sides to get . Divide every term by to isolate : . Thus, Option 1 is correct. Option 2 reverses subtraction, Option 3 does not account for division, and Option 4 misconstrues the equation structure. Following the algebraic manipulations accurately results in the correct expression.

Systems of Equations in Two Variables Question 12:

A car rental company charges a flat fee of plus per mile driven. If the total cost is , how many miles were driven?

  1. 320 miles
  2. 3200 miles
  3. 200 miles
  4. 400 miles

Answer (Detailed Solution Below)

Option 1 : 320 miles

Systems of Equations in Two Variables Question 12 Detailed Solution

The cost equation given by the rental company is , where is the number of miles driven. To find , first subtract from both sides: . Then, divide both sides by to solve for : . Therefore, the number of miles driven is 320, making option 1 the correct answer. Other options do not satisfy the equation when plugged in.

Systems of Equations in Two Variables Question 13:

Find the value of for which the system and is satisfied.

  1. 0
  2. 1
  3. 2
  4. 3

Answer (Detailed Solution Below)

Option 4 : 3

Systems of Equations in Two Variables Question 13 Detailed Solution

Express from as . Substitute into : . Simplify to , rearranging to . Solve using the quadratic formula: , where , , . Calculate . Thus, or . Verify satisfies both equations. Therefore, is correct.

Systems of Equations in Two Variables Question 14:

Given the equations and , find the value of for the solution to this system.

  1. 1
  2. 3
  3. -1
  4. -3

Answer (Detailed Solution Below)

Option 2 : 3

Systems of Equations in Two Variables Question 14 Detailed Solution

To solve the system and , we start by expressing in terms of from the second equation: . Substitute this expression for in the first equation: . Simplifying gives . Rearrange to form . Factor the quadratic to get . Thus, the solutions for are and . Only is valid when substituted back into and checked with . Thus, the correct answer is .

Systems of Equations in Two Variables Question 15:

Find the value of if and .

  1. 2
  2. 4
  3. -2
  4. -4

Answer (Detailed Solution Below)

Option 1 : 2

Systems of Equations in Two Variables Question 15 Detailed Solution

From the equations and , substitute into the first equation: . Simplify to . Rearrange: . Solve using the quadratic formula , where , , . Thus, . Simplify to or . Checking both solutions, only satisfies both original equations. Therefore, is correct.

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