Linear Equations and Their Properties

The video tutorial reviews linear equations in two variables, explaining their general form as ax + by + c = 0, where a, b, and c are constants. It emphasizes that both a and b cannot be zero simultaneously. The tutorial also demonstrates how to find solutions to these equations by substituting values for x and y that satisfy the equation. Examples are provided to illustrate the process of verifying solutions.

Which of the following is an example of a linear equation in two variables?

• x^2 + y^2 = 25

• 2x + 5y = 12

• 3x = 9

• y = 4

Which of the following is a valid linear equation in two variables?

• 2x^2 + 3y = 6

• x + y^2 = 5

• 3x + 4y = 7

• x^2 + y^2 = 1

What is the general form of a linear equation in two variables?

• ax^2 + by^2 + c = 0

• ax^2 + by = c

• ax + by + c = 0

• ax + by = c

In the equation ax + by + c = 0, what are 'a' and 'b' referred to as?

• Variables

• Solutions

• Coefficients

• Constants

Which condition must be true for the coefficients 'a' and 'b' in a linear equation?

• a and b must both be non-zero

• a and b must be equal

• a and b can both be zero

• At least one of a or b must be non-zero

How can the condition that 'a' and 'b' cannot both be zero be mathematically expressed?

• a + b ≠ 0

• a^2 - b^2 ≠ 0

• a^2 + b^2 ≠ 0

• a * b ≠ 0

What is the role of 'c' in the equation ax + by + c = 0?

• Coefficient

• Solution

• Constant

• Variable

What happens if both 'a' and 'b' are zero in the equation ax + by + c = 0?

• The equation represents a line

• The equation becomes a constant

• The equation represents a point

• The equation becomes undefined

What is a solution to a linear equation in two variables?

• A constant value

• A pair of values for x and y

• A single value for y

• A single value for x

For the equation 2x + 5y = 12, which of the following pairs is a solution?

• (2, 3)

• (0, 0)

• (3, 1)

• (1, 2)