Linear Equations In Two Variables - 1 - Free MCQ Practice Test with solutions,

The given equation x = 5 is a linear equation in one variable. To write it as a linear equation in two variables (x and y), we can add 0.y since it does not affect the value of the equation.
Thus, the equation becomes:
x + 0y - 5 = 0
This matches option D.

A: x + 5 = 6 is a linear equation in one variable because it can be rewritten as x = 1, which involves only one variable, x, and its highest power is 1.
B: 2x + 3y = 0 is a linear equation in two variables because it involves two variables, x and y.
C: x² = 5x + 3 is a quadratic equation in one variable because the highest power of x is 2.
D: 5x = y² + 3 is not a linear equation because it involves a variable, y, raised to the power of 2.

For the equation ax + by + c = 0 to be a linear equation, the following must hold:
• The equation should represent a straight line.
• At least one of the coefficients of x (a) or y (b) must be non-zero to ensure that it involves either x, y, or both.
If both a = 0 and b = 0, the equation reduces to c = 0, which is not a linear equation as it does not involve x or y.
Thus, a and b shouldn't be zero at the same time for the equation to be linear.

The linear equation (2x + 3y = 6) can be rewritten in the slope-intercept form (y = mx + b).
- Rearrange to (3y = -2x + 6) and then (y = - 2/3x + 2).
- This is a straight-line equation.
- A straight line has infinitely many points.
Thus, the equation has:
- Infinitely many solutions (C)

A linear equation in two variables has maximum:
C: Infinite solutions
- A linear equation in two variables can be written as ( ax + by = c ).
- This equation represents a straight line on the coordinate plane.
- Every point on this line is a solution to the equation.
- Since there are infinitely many points on a line, there are infinitely many solutions to the equation.
Hence, the correct answer is C: Infinite solutions.

To find if ( x = 2, y = -1 ) is a solution for a given equation, substitute these values into each equation:
- A: ( 2(2) + 3(-1) =1 not  5 ) ✗ 
- B: ( 2 + (-1) = 1 not 5 ) ✗ 
- C: ( 2 + (-1) = 1 ) ✓
- D: ( 2 - (-1) = 3 not 9 ) ✗ 
Therefore, option C is the correct answer.

Correct Answer: (iii) Infinitely many solutions
x + y = 10 is a linear equation in two variables, hence it has infinitely many solutions.

Transposing 6 from RHS to LHS we get 
x + 2y − 6 = 0

Therefore, Answer: (ii) x + 2y − 6 = 0 

The given equation is 7 = 3x. Rearranging it into the standard form ax + by + c = 0, we get
3x - 7 = 0.
Here, a = 3, b = 0, and c = -7.
Thus, the solution is a = 3, b = 0, c = -7.

Option (a):

2x + 3 = 7x – 2

• Rearrange: 2x + 3 = 7x – 2
• Here, x appears only with power 1.

This is a linear equation.

Option (b):

(2/3)x + 5 = 3x – 4

• Coefficient 2/3 is just a number; it doesn’t affect the power.
• Variable x is still to the power 1.

This is a linear equation.

Option (c):

x² + 3 = 5x – 3

• Rearrange: x² – 5x + 6 = 0
• Here, the highest power of x is 2.

This is not a linear equation (this is a quadratic equation).

Option (d):

(x – 2)² = x² + 8

• Expand LHS: x² – 4x + 4 = x² + 8
• The x² terms cancel, leaving: –4x + 4 = 8 → –4x = 4 → x = –1
• After simplification, it becomes a linear equation.

This is a linear equation (even though it looked quadratic at first).