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Linear Equations in Two Variables - Equations of Lines Parallel to the x-axis and y-axis

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A linear equation in two variables is typically expressed as ax+by+c=0ax + by + c = 0. When one of the coefficients (aa or bb) is zero, the resulting line is parallel to one of the coordinate axes.

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The equation of the x-axis is y=0y = 0. Visually, this is the horizontal line where every point has a y-coordinate of zero, regardless of the value of xx, such as (2,0)(2, 0), (−3,0)(-3, 0), or (0,0)(0, 0).

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The equation of the y-axis is x=0x = 0. Visually, this is the vertical line where every point has an x-coordinate of zero, regardless of the value of yy, such as (0,5)(0, 5), (0,−2)(0, -2), or (0,0)(0, 0).

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An equation of the form y=ky = k (where kk is a constant) represents a line parallel to the x-axis. Visually, this is a horizontal line that intersects the y-axis at (0,k)(0, k). If k>0k > 0, the line lies above the x-axis; if k<0k < 0, it lies below it.

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An equation of the form x=kx = k (where kk is a constant) represents a line parallel to the y-axis. Visually, this is a vertical line that intersects the x-axis at (k,0)(k, 0). If k>0k > 0, the line is to the right of the y-axis; if k<0k < 0, it is to the left.

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An equation in one variable, such as x=3x = 3, can be treated as a linear equation in two variables by writing it as 1x+0y=31x + 0y = 3. This shows that for any real value of yy, xx will always be 33, resulting in a straight vertical line on a graph.

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The intersection of a line parallel to the x-axis (y=by = b) and a line parallel to the y-axis (x=ax = a) is always the point (a,b)(a, b). Visually, these two perpendicular lines meet at a single unique coordinate point.

📐Formulae

General linear equation: ax+by+c=0ax + by + c = 0

Equation of the x-axis: y=0y = 0

Equation of the y-axis: x=0x = 0

Line parallel to the x-axis: y=ky = k (or 0x+1y=k0x + 1y = k)

Line parallel to the y-axis: x=kx = k (or 1x+0y=k1x + 0y = k)

💡Examples

Problem 1:

Solve the equation 2x+10=02x + 10 = 0 and represent the solution on (i) the number line and (ii) the Cartesian plane.

Solution:

Step 1: Solve for xx in the equation 2x+10=02x + 10 = 0. 2x=−102x = -10 x=−102x = \frac{-10}{2} x=−5x = -5 Step 2: On the number line, the solution x=−5x = -5 is a single point located 5 units to the left of zero. Step 3: On the Cartesian plane, we express it as 1x+0y=−51x + 0y = -5. This is a vertical line passing through the point (−5,0)(-5, 0), parallel to the y-axis.

Explanation:

While x=−5x = -5 is a single point in one dimension, it represents an infinite set of points in two dimensions where the x-coordinate is fixed but the y-coordinate can be any value.

Problem 2:

Find the equation of a line parallel to the x-axis that passes through the point (3,−4)(3, -4).

Solution:

Step 1: Identify the form of the equation. A line parallel to the x-axis is horizontal and has the form y=ky = k. Step 2: Use the given point (3,−4)(3, -4). Since the line passes through this point, the y-coordinate of the point must satisfy the equation. Step 3: Here, the y-coordinate is −4-4, so k=−4k = -4. Step 4: The equation of the line is y=−4y = -4 or y+4=0y + 4 = 0.

Explanation:

Because the line is parallel to the x-axis, the x-coordinate of the point (3)(3) does not affect the equation. Only the y-coordinate determines the height of the horizontal line.