GSEB Class 9 Maths Solutions Chapter 4 Linear Equations in Two Variables Exercise 4.4

Question 1. Give the geometric representations of \( y = 3 \) as an equation
(i) in one variable.
(ii) in two variables.
Answer:
(i) The equation provided is \( y = 3 \). For one variable, \( y = 3 \) shows a specific point. When we represent \( y = 3 \) on a number line, it just marks one location. To do this, we draw a vertical number line and find that exact spot at 3.
(ii) In two variables, the equation \( y = 3 \) can be written as \( 0x + 1y = 3 \). This forms a linear equation for variables \( x \) and \( y \), which we show as a straight line. All values for \( x \) are allowed since \( 0 \cdot x \) is always \( 0 \). However, \( y \) must always be \( 3 \). For example, two solutions for this equation are \( (0, 3) \) and \( (2, 3) \). Consequently, the graph named AB creates a line that runs parallel to the x-axis, located 3 units above the starting point (origin).
In simple words: When \( y = 3 \) is shown with only one variable, it means to mark the point 3 on a number line. If it's a 'y' variable, you can imagine it on a vertical line. For two variables, \( y = 3 \) means a straight line where 'y' is always 3, no matter what 'x' is. This line runs flat, 3 steps up from the center, parallel to the 'x' line.

Exam Tip: Remember that an equation in one variable, like \( y = 3 \), always represents a single point on a number line or a single line parallel to an axis in a 2D plane. An equation like \( y = k \) (where k is a constant) in two variables always represents a line parallel to the x-axis, passing through \( (0, k) \).

Question 2. Give the geometric representations of \( 2x + 9 = 0 \) as an equation
(i) in one variable (ii) in two variables
Answer:
(i) The given equation is \( 2x + 9 = 0 \). We can find the value of \( x \) as \( 2x = -9 \), so \( x = -\frac{9}{2} = -4.5 \). Representing \( x = -4.5 \) on a number line displays a specific point. We simply mark this location on the line.
(ii) For two variables, the equation \( 2x + 9 = 0 \) can be written as \( 2x + 0y + 9 = 0 \). This represents a linear equation with variables \( x \) and \( y \), shown as a straight line. All values for \( y \) are allowed because \( 0 \cdot y \) is always \( 0 \). However, \( x \) must satisfy \( 2x + 9 = 0 \), meaning \( x = -\frac{9}{2} \). Two solutions for this equation are \( (-\frac{9}{2}, 0) \) and \( (-\frac{9}{2}, 2) \). Therefore, the graph of AB forms a line that is parallel to the y-axis, located \( 4.5 \) units to the left of the starting point (origin).
In simple words: To show \( 2x + 9 = 0 \) with one variable, first solve for \( x \), which is \( -4.5 \). Then, simply mark this spot on a number line. For two variables, \( 2x + 9 = 0 \) means a straight line where 'x' is always \( -4.5 \), no matter what 'y' is. This line runs up and down, \( 4.5 \) steps to the left of the center, parallel to the 'y' line.

Exam Tip: Always simplify the equation to the form \( x = k \) or \( y = k \) before attempting to represent it geometrically in one variable. An equation like \( x = k \) (where k is a constant) in two variables always represents a line parallel to the y-axis, passing through \( (k, 0) \).