krit.club logo

Linear Equations in Two Variables - Plot and interpret graphs of linear equations on the Cartesian plane

Grade 9CBSE

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

🔑Concepts

•

A linear equation in two variables ax+by+c=0ax + by + c = 0 is represented geometrically as a straight line on the Cartesian plane. Every point (x,y)(x, y) that satisfies the equation lies on this line, and every point on the line is a solution to the equation.

Graph of a linear equation showing a straight line passing through specific points on a Cartesian plane.
•

To plot the graph, we typically find at least two solutions (ordered pairs) of the equation. For example, by setting x=0x = 0 we find the yy-intercept, and by setting y=0y = 0 we find the xx-intercept. Connecting these points with a straight line defines the graph.

Plotting a line using x and y intercepts.
•

Equations of the form x=kx = k represent vertical lines parallel to the yy-axis, while equations of the form y=ky = k represent horizontal lines parallel to the xx-axis.

Vertical line x=3 and horizontal line y=-2.
•

The graph of y=mxy = mx always passes through the origin (0,0)(0, 0). This represents a direct variation between xx and yy.

📐Formulae

Standard Form: ax+by+c=0ax + by + c = 0

Slope-Intercept Form: y=mx+cy = mx + c

Isolated variable form for calculation: y=frac−(ax+c)by = \\frac{-(ax + c)}{b}

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

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

💡Examples

Problem 1:

Find three different solutions for the equation 2x+3y=122x + 3y = 12.

Solution:

Step 1: Let x=0x = 0. Substituting into the equation: 2(0)+3y=12Rightarrow3y=12Rightarrowy=42(0) + 3y = 12 \\Rightarrow 3y = 12 \\Rightarrow y = 4. So, (0,4)(0, 4) is a solution. \nStep 2: Let y=0y = 0. Substituting into the equation: 2x+3(0)=12Rightarrow2x=12Rightarrowx=62x + 3(0) = 12 \\Rightarrow 2x = 12 \\Rightarrow x = 6. So, (6,0)(6, 0) is a solution. \nStep 3: Let x=3x = 3. Substituting into the equation: 2(3)+3y=12Rightarrow6+3y=12Rightarrow3y=6Rightarrowy=22(3) + 3y = 12 \\Rightarrow 6 + 3y = 12 \\Rightarrow 3y = 6 \\Rightarrow y = 2. So, (3,2)(3, 2) is a solution.

Explanation:

To find solutions, we arbitrarily chose values for one variable and solved for the other. Each pair (x,y)(x, y) represents a point on the graph of the line 2x+3y=122x + 3y = 12.

Problem 2:

Check if the point (2,−2)(2, -2) is a solution to the equation x−2y=6x - 2y = 6.

Solution:

Step 1: Identify the coordinates x=2x = 2 and y=−2y = -2. \nStep 2: Substitute these values into the LHS of the equation: LHS=x−2y=2−2(−2)LHS = x - 2y = 2 - 2(-2). \nStep 3: Simplify the expression: 2+4=62 + 4 = 6. \nStep 4: Compare LHS and RHS. Since LHS=6LHS = 6 and RHS=6RHS = 6, LHS=RHSLHS = RHS.

Explanation:

Because the substitution makes the equation true, the ordered pair (2,−2)(2, -2) is a solution, meaning this point lies exactly on the line x−2y=6x - 2y = 6.

Problem 3:

Draw the graph of the linear equation x+y=4x + y = 4.

Graph of x + y = 4 passing through (0,4), (4,0) and (2,2).

Solution:

  1. Find solutions: If x=0x = 0, 0+y=4  ⟹  y=40 + y = 4 \implies y = 4. Point is (0,4)(0, 4). If y=0y = 0, x+0=4  ⟹  x=4x + 0 = 4 \implies x = 4. Point is (4,0)(4, 0). If x=2x = 2, 2+y=4  ⟹  y=22 + y = 4 \implies y = 2. Point is (2,2)(2, 2).
  2. Plot these points (0,4)(0, 4), (4,0)(4, 0), and (2,2)(2, 2) on the graph.
  3. Draw a line passing through these points.

Explanation:

To graph a linear equation, we find coordinate pairs that make the equation true. The intercepts (where the line crosses the axes) are usually the easiest points to calculate.

Problem 4:

Draw the graph of the linear equation 2x−y=32x - y = 3. Find the coordinates of the point where the graph cuts the yy-axis.

Graph of 2x - y = 3 showing points A(0,-3) and B(2,1).

Solution:

Step 1: Express yy in terms of xx: y=2x−3y = 2x - 3. Step 2: Find at least two solutions. If x=0x = 0, y=2(0)−3=−3y = 2(0) - 3 = -3. Point is A(0,−3)A(0, -3). If x=2x = 2, y=2(2)−3=1y = 2(2) - 3 = 1. Point is B(2,1)B(2, 1). Step 3: Plot points A(0,−3)A(0, -3) and B(2,1)B(2, 1) on the Cartesian plane and join them with a straight line. Step 4: Observation: The line cuts the yy-axis at (0,−3)(0, -3).

Explanation:

To plot the graph, we calculate coordinates that satisfy the equation. The yy-axis intersection occurs where the xx-coordinate is zero.

Plot and interpret graphs of linear equations on the Cartesian plane Class 9 Notes & Examples