Graphical Solution of a Pair of Linear Equations

Video Explanation

Question

Solve graphically the following system of linear equations. Also find the coordinates of the points where the lines meet the x-axis:

\[ x + 2y = 5 \]

\[ 2x – 3y = -4 \]

Solution

Step 1: Write Both Equations in the Form \(y = mx + c\)

Equation (1):

\[ x + 2y = 5 \Rightarrow 2y = 5 – x \Rightarrow y = \frac{5 – x}{2} \]

Equation (2):

\[ 2x – 3y = -4 \Rightarrow 3y = 2x + 4 \Rightarrow y = \frac{2x + 4}{3} \]

Step 2: Prepare Tables of Values

For Equation (1): \(y = \frac{5 – x}{2}\)

x	y
1	2
5	0

For Equation (2): \(y = \frac{2x + 4}{3}\)

x	y
1	2
-2	0

Step 3: Graphical Representation

Plot the following points on the same Cartesian plane:

• Line 1: (1, 2) and (5, 0)
• Line 2: (1, 2) and (−2, 0)

Join each pair of points to obtain two straight lines.

The two straight lines intersect at the point (1, 2).

Result

The graphical solution of the given system of equations is:

\[ x = 1,\quad y = 2 \]

Points Where the Lines Meet the X-Axis

For equation (1), when \(y = 0\):

\[ x = 5 \Rightarrow \text{Point} = (5, 0) \]

For equation (2), when \(y = 0\):

\[ 2x = -4 \Rightarrow x = -2 \Rightarrow \text{Point} = (-2, 0) \]

Conclusion

The given system of linear equations has a unique solution at the point (1, 2).

The points where the lines meet the x-axis are:

• (5, 0)
• (−2, 0)