NCERT Solutions & Concepts for Class 9 Maths Linear Equation in Two Variables Ex 4.4

Geometric Representation: 1D vs 2D Coordinate Systems

A linear equation can be visualized in different dimensional spaces. When we consider an equation in one variable (such as $y = c$ or $x = c$), we are working in a single dimension: the standard real number line. In this 1D space, the equation represents a unique single point. However, when we express the exact same relationship in two variables (by introducing the missing variable with a zero coefficient, such as $0x + y = c$), we expand our visualization into the 2D Cartesian plane. Here, the equation describes an infinite locus of points forming a straight line. Specifically, $y = c$ represents a horizontal line parallel to the x-axis, while $x = c$ represents a vertical line parallel to the y-axis.

Steps to Represent Equations Geometrically

1. Simplify the Equation — Solve the given linear equation for the variable. For example, simplify the equation $2x + 9 = 0$ to get $x = -4.5$.
2. Plot in One Variable (Number Line) — Draw a horizontal real number line. Mark the origin (0) and scale it. Locate the single numerical value (e.g., $-4.5$) and mark it clearly with a solid dot.
3. Convert to Two-Variable Form — Express the simplified equation in the standard form $ax + by + c = 0$. For $x = -4.5$, write it as $1x + 0y = -4.5$. For $y = 3$, write it as $0x + 1y = 3$.
4. Plot in Two Variables (Cartesian Plane) — Draw the X and Y axes. If $x$ is constant, draw a vertical line passing through $(x, 0)$ parallel to the Y-axis. If $y$ is constant, draw a horizontal line passing through $(0, y)$ parallel to the X-axis.

Step-by-Step Solved NCERT Examples

• Example 1: Give the geometric representation of $y = 3$ as an equation. 1D Representation (One Variable): The equation is already solved. On a single horizontal number line, mark a point at position $+3$. 2D Representation (Two Variables): Write it as $0 \cdot x + y = 3$. Pick arbitrary values for $x$ to find points: if $x=0$, $y=3$; if $x=2$, $y=3$. Plot $(0, 3)$ and $(2, 3)$ on the Cartesian plane. Join them to form a straight horizontal line parallel to the x-axis passing through $(0, 3)$.
• Example 2: Give the geometric representation of $2x + 9 = 0$ as an equation. 1D Representation (One Variable): Solve $2x = -9 \implies x = -4.5$. On a horizontal number line, locate and mark the point $-4.5$ exactly halfway between $-4$ and $-5$. 2D Representation (Two Variables): Write it as $2x + 0 \cdot y + 9 = 0$. Here, $x$ is fixed at $-4.5$ while $y$ can be any real number. Plot points like $(-4.5, 0)$, $(-4.5, 2)$, and $(-4.5, -2)$. Draw a straight vertical line passing through these points, which is parallel to the y-axis.

Crucial Exam Tips & Common Pitfalls

• The Zero Coefficient Rule: When converting to two variables, do not omit the missing variable. Always represent it with a 0 coefficient (e.g., $x = 3 \implies 1x + 0y = 3$).
• Parallel Line Confusion: Remember that if $y$ is constant ($y = k$), the line is parallel to the opposite axis (the X-axis). If $x$ is constant ($x = k$), the line is parallel to the Y-axis.
• Labeling Requirements: In CBSE board papers, always label your coordinate axes ($X, X', Y, Y'$), identify the origin $(0,0)$, specify the scale used, and write the algebraic equation along the drawn line to secure full marks.

Practice Questions with Solutions

• Q: Give the geometric representation of $3x - 6 = 0$ as an equation in (i) one variable, and (ii) two variables. A: Step 1: Simplify the given equation $3x - 6 = 0$. $3x = 6 \implies x = 2$. Step 2: Represent in one variable. On a horizontal number line, plot a solid point at the coordinate position $2$. Step 3: Represent in two variables. Rewrite the equation as $1x + 0y = 2$. Since $y$ can be any value, pick coordinate points like $(2, 0)$, $(2, 1)$, and $(2, -2)$. Plot these points on a Cartesian plane. Step 4: Draw the line. Draw a vertical straight line passing through $(2,0)$. This line is parallel to the y-axis. Final answer: One variable is a point at $2$ on the number line; two variables is a vertical line $x=2$ parallel to the y-axis.
• Q: Show that the line representing the equation $2y + 8 = 0$ is parallel to the x-axis when plotted on a Cartesian plane. A: Step 1: Solve the equation $2y + 8 = 0$ for $y$. $2y = -8 \implies y = -4$. Step 2: Convert to two-variable form. The equation becomes $0x + y = -4$. Step 3: Generate coordinates. Since the coefficient of $x$ is $0$, $y$ remains $-4$ for any real value of $x$. Valid points include $(0, -4)$, $(2, -4)$, and $(-2, -4)$. Step 4: Connect the points. Plotted coordinates form a straight horizontal line situated 4 units below the origin. Final answer: Since all points on this line maintain a constant distance of 4 units below the x-axis, the line is parallel to the x-axis.
• Q: Write the equation of a line passing through the point $(4, -7)$ which is parallel to the x-axis. A: Step 1: Recall the general equation form. A line parallel to the x-axis has a constant y-coordinate and is represented by $y = k$. Step 2: Find the value of $k$ from the given point. The line passes through $(4, -7)$ where the y-coordinate is $-7$. Therefore, $k = -7$. Step 3: Write the final equation. The equation is $y = -7$, which can also be written in two variables as $0x + 1y = -7$. Final answer: The equation is $y = -7$.
• Q: Determine whether the coordinate point $(-3, 5)$ lies on the line representing the equation $4x + 12 = 0$ in two variables. A: Step 1: Simplify the equation $4x + 12 = 0$. $4x = -12 \implies x = -3$. Step 2: Express in two-variable form. $1x + 0y = -3$. Step 3: Substitute the coordinates $(-3, 5)$ into the equation. Substitute $x = -3$ and $y = 5$: $1(-3) + 0(5) = -3 \implies -3 = -3$. Step 4: Analyze the statement. Since LHS equals RHS, the point satisfies the equation. Final answer: Yes, the point $(-3, 5)$ lies on the line $4x + 12 = 0$.

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