krit.club logo

Linear Equations in Two Variables - Graph of a Linear Equation in Two Variables

Grade 9CBSE

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

🔑Concepts

•

Every linear equation in two variables of the form ax+by+c=0ax + by + c = 0 represents a straight line when plotted on a Cartesian plane. Visually, this means that all the points (x,y)(x, y) that satisfy the equation lie on this single straight path, and any point not on the line does not satisfy the equation.

•

To plot the graph of a linear equation, you need to find at least two solutions (coordinates). Visually, these are points (x,y)(x, y) on the grid. While two points are enough to define a line, finding a third point is recommended to ensure they all align in a straight row, which confirms your calculations are correct.

•

The horizontal axis is the xx-axis, and its equation is y=0y = 0. The vertical axis is the yy-axis, and its equation is x=0x = 0. Any point lying on the xx-axis will have a vertical position of zero, and any point on the yy-axis will have a horizontal position of zero.

•

A linear equation of the form y=mxy = mx always passes through the origin (0,0)(0, 0). Visually, this line will cross the exact center where the xx-axis and yy-axis intersect.

•

An equation of the form x=ax = a results in a vertical line parallel to the yy-axis. For example, x=3x = 3 is a straight vertical line passing through 33 on the xx-axis. Visually, every point on this line has the same horizontal distance from the yy-axis.

•

An equation of the form y=by = b results in a horizontal line parallel to the xx-axis. For example, y=−2y = -2 is a straight horizontal line passing through −2-2 on the yy-axis. Visually, every point on this line stays at the same vertical 'height' relative to the xx-axis.

•

A linear equation in two variables has infinitely many solutions, which translates to the fact that the line extending from the points can be drawn infinitely in both directions. In a graph, this is often indicated by drawing arrows at both ends of the line.

📐Formulae

General form of a linear equation in two variables: ax+by+c=0ax + by + c = 0

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

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

Equation of a line parallel to the yy-axis: x=ax = a

Equation of a line parallel to the xx-axis: y=by = b

Equation of a line passing through the origin: y=mxy = mx

💡Examples

Problem 1:

Draw the graph of the linear equation x+2y=6x + 2y = 6.

Solution:

  1. Find at least two solutions for the equation.
    • Let x=0x = 0: 0+2y=6  ⟹  2y=6  ⟹  y=30 + 2y = 6 \implies 2y = 6 \implies y = 3. Point is A(0,3)A(0, 3).
    • Let y=0y = 0: x+2(0)=6  ⟹  x=6x + 2(0) = 6 \implies x = 6. Point is B(6,0)B(6, 0).
    • Let x=2x = 2: 2+2y=6  ⟹  2y=4  ⟹  y=22 + 2y = 6 \implies 2y = 4 \implies y = 2. Point is C(2,2)C(2, 2).
  2. Plot the points (0,3)(0, 3), (6,0)(6, 0), and (2,2)(2, 2) on a Cartesian plane.
  3. Use a ruler to draw a straight line passing through these points and extend it with arrows at both ends.

Explanation:

This approach uses the intercept method (setting x=0x=0 and y=0y=0) to find where the line crosses the axes, which is the most efficient way to graph a linear equation.

Problem 2:

Check whether the point (2,−2)(2, -2) lies on the graph of the equation 3x+4y=−23x + 4y = -2.

Solution:

  1. Substitute x=2x = 2 and y=−2y = -2 into the Left Hand Side (LHS) of the equation 3x+4y=−23x + 4y = -2.
  2. LHS=3(2)+4(−2)=6−8=−2LHS = 3(2) + 4(-2) = 6 - 8 = -2.
  3. RHS=−2RHS = -2.
  4. Since LHS=RHSLHS = RHS, the point (2,−2)(2, -2) satisfies the equation.

Explanation:

If a point satisfies the algebraic equation of the line, it must geometrically lie on the line when represented on a graph. Since the calculation holds true, the point (2,−2)(2, -2) is a part of the line's graph.