Linear Equation In Two Variables - Olympiad Level MCQ, Class 9 Mathematics - Class 9 MCQ

Let b be Rozly's rowing speed in still water and c be the speed of the current.

1. Downstream:

• Distance = 20 km
• Time = 2 hours
• Speed downstream = 20/2 = 10 km/hr
• Equation: b + c = 10

2. Upstream:

• Distance = 4 km
• Time = 2 hours
• Speed upstream = 4/2 = 2 km/hr
• Equation: b - c = 2

3. Solving the equations:

• Adding both equations:
• (b + c) + (b - c) = 10 + 2
• 2b = 12
• b = 6 km/hr

Thus, Rozly's rowing speed in still water is 6 km/hr.

Given the equations:

• 1. 2a - 3 = 5
  • Add 3 to both sides: 2a = 8
  • Divide by 2: a = 4
• 2. 3b + 1 = 2
  • Subtract 1 from both sides: 3b = 1
  • Divide by 3: b = ¹/₃

Now, compute 3b - 2a:

• 3b = 3 × ¹/₃ = 1
• 2a = 2 × 4 = 8

Thus, 3b - 2a = 1 - 8 = -7.

The correct answer is -7, which corresponds to option B.

To determine where the line y = 2x + 3 intersects the y-axis, we follow these steps:

• Set x to 0.
• Solve for y:
• y = 2(0) + 3
• y = 3

Therefore, the line crosses the y-axis at the point (0, 3).

To determine which line the point (2, 1) lies on, substitute x = 2 and y = 1 into each equation:

• Option A: x = y
  • 2 = 1 is false.
• Option B: y = x + 1
  • 1 = 2 + 1 simplifies to 1 = 3, which is false.
• Option C: 2y = x
  • 2(1) = 2 simplifies to 2 = 2, which is true.
• Option D: xy = 1
  • 2 × 1 = 2 ≠ 1, so it's false.

Thus, the correct answer is option C.

To determine which ordered pair lies on the line y = 2x - 1, substitute each point into the equation.

• For option A:
  • x = 1
  • y = 2(1) - 1 = 1
  • The point (1, 1) satisfies the equation.
• For option B:
  • x = 2
  • y = 2(2) - 1 = 3
  • The point (2, 3) satisfies the equation.
• For option C:
  • x = 0
  • y = 2(0) - 1 = -1
  • The point (0, -1) satisfies the equation.
• For option D:
  • x = -1
  • y = 2(-1) - 1 = -3
  • The point (-1, -3) satisfies the equation.

All options now correctly satisfy the equation.

To determine which ordered pairs satisfy the inequality x + y + 1 < 0, we substitute each pair into the inequality:

• For option A: 0 + (-1) + 1 = 0, which does not satisfy 0 < 0.
• For option B: -2 + 0 + 1 = -1, which satisfies -1 < 0.
• For option C: 2 + (-4) + 1 = -1, which satisfies -1 < 0.

Since both options B and C satisfy the inequality, the correct answer is D.

The equation of the line is given in slope-intercept form, which is represented as:

y = mx + b

• y is the dependent variable.
• m represents the slope of the line.
• x is the independent variable.
• b is the y-intercept, the point where the line crosses the y-axis.

In this case, the coefficient of x is 2. Therefore, the slope of the line is:

m = 2

To determine which pair of equations has the same solutions, solve each equation for x.

• For option A:
  • 6x = 3 implies x = 3/6 = 0.5
  • 3x = 1.5 implies x = 1.5/3 = 0.5
  • Both equations have the same solution, x = 0.5.
• For option B:
  • 8x = 3 implies x = 3/8 = 0.375
  • 4x = 1 implies x = 1/4 = 0.25
  • Different solutions.
• For option C:
  • 10x = 9 implies x = 9/10 = 0.9
  • 5x = 18 implies x = 18/5 = 3.6
  • Different solutions.
• For option D:
  • 12x = 6 implies x = 6/12 = 0.5
  • 6x = 6 implies x = 6/6 = 1
  • Different solutions.

Only option A has equations with the same solution.

To determine which option is not a solution to the equation 2x + 3y = 6, we substitute each pair into the equation:

• For option A (3, 0):
  2(3) + 3(0) = 6 + 0 = 6
  This satisfies the equation.
• For option B (0, 2):
  2(0) + 3(2) = 0 + 6 = 6
  This also satisfies the equation.
• For option C (-3, 4):
  2(-3) + 3(4) = -6 + 12 = 6
  This satisfies the equation as well.
• For option D (1, 1):
  2(1) + 3(1) = 2 + 3 = 5
  This does not equal 6.

Therefore, the correct answer is D.

The general form of the equation of a line is ax + by + c = 0, where at least one of a or b must be non-zero for it to represent a unique line.

• Option A: If a = 0, c = 0 and b ≠ 0, the equation becomes by = 0 → y = 0, which is a valid line (the x-axis).
• Option B: If b = 0, c = 0 and a ≠ 0, the equation becomes ax = 0 → x = 0, which is also a valid line (the y-axis).
• Option C: If a = 0 and b = 0, the equation reduces to c = 0. This does not represent any specific line; it either represents all points in the plane if c = 0 or no graph if c ≠ 0.
• Option D: If c = 0 and both a and b are non-zero, the equation becomes ax + by = 0, which is a valid line passing through the origin.

Thus, the correct answer is Option C because when both coefficients of x and y are zero (a = 0 and b = 0), the equation does not represent a specific line.

The given equation is y = 3x + 5n. This equation involves three variables: x, y, and n.

In a system of equations, the number of solutions depends on the number of equations available. Here, we only have one equation with three variables, which means we cannot find unique values for all three variables. In such cases, the solution set is infinite because:

• You can choose arbitrary values for two variables (e.g., x and n).
• The third variable (y) will be determined based on those choices.

Therefore, this equation has infinitely many solutions.

If it was intended to have only two variables (for example, if n was a typo), the analysis would still hold because:

• One equation with two variables also results in infinitely many solutions.

Thus, the correct answer is D: Infinitely many solutions.

Let the missing member be x. According to the problem, the second member of the pair is 4 more than the first member. This can be expressed as:

-8 = x + 4.

To find x, subtract 4 from both sides:

x = -8 - 4.

Therefore, x = -12.

Thus, the missing member is -12, which corresponds to option C.

To find the point of intersection of the lines x + y = 1 and x - y = 1, we solve the equations simultaneously.

• Step 1: Add the two equations to eliminate y:
  (x + y) + (x - y) = 1 + 1
  Simplifying:
  2x = 2 implies x = 1
• Step 2: Substitute x = 1 into the first equation to find y:
  1 + y = 1 implies y = 0

Thus, the point of intersection is (1, 0), which corresponds to option B.

To determine the point of intersection of the lines x + y = 1 and 2x + 2y = 4, we can analyse their equations:

• The first line is given by: x + y = 1
• The second line simplifies as follows: 2x + 2y = 4 implies x + y = 2 (Dividing both sides of the equation by 2)

Now, we have two equations:

• x + y = 1 (1)
• x + y = 2 (2)

Subtracting equation (1) from equation (2):

• (x + y) - (x + y) = 2 - 1
• 0 = 1

This result is a contradiction (0 = 1), which indicates that the system of equations has no solution. Therefore, the lines are parallel and do not intersect at any point.

Hence, the correct answer is C. No intersection point.

The cost of a notebook (x) is twice the cost of a pen (y), leading to the equation:

x = 2y

Rearranging this gives:

x - 2y = 0

This corresponds to option B.

The equation x = 4 represents a vertical line where:

• Every point on the line has an x-coordinate of 4.
• The y-coordinate can be any real number.

Among the given options, only point A (4, 3) has an x-coordinate of 4. Therefore, the correct answer is A.

The equation y = 3 represents a horizontal line that passes through all points where the y-coordinate is 3. Among the given options:

• Only option C has a y-coordinate of 3.

Therefore, the correct answer is C.

To determine which option is not a solution to the equation 3x + 4y = 12, we substitute each pair into the equation:

• Option A: 3(8) + 4(-3) = 24 - 12 = 12 ✓
• Option B: 3(0) + 4(3) = 0 + 12 = 12 ✓
• Option C: 3(2) + 4(3) = 6 + 12 = 18 ≠ 12 ✗
• Option D: 3(4) + 4(0) = 12 + 0 = 12 ✓

Since option C does not satisfy the equation, the correct answer is C.

On the Cartesian plane, any point on the x-axis has a y-coordinate of zero. Therefore, such points are represented as (x, 0). This distinguishes them from points on the y-axis or general points in the plane.

Thus, the correct form is option A, (x, 0).

A point on the y-axis has an x-coordinate of 0 and can have any y-value. Therefore, it should be in the form (0, y).

• Option A is incorrect because it uses (0, x), which swaps the variables.
• Option C is incorrect as it represents a point on the x-axis.

Thus, none of the provided options correctly represent a point on the y-axis.