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 y-axis:

\[ 3x + y – 5 = 0 \]

\[ 2x – y – 5 = 0 \]

Solution

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

Equation (1):

\[ 3x + y – 5 = 0 \Rightarrow y = 5 – 3x \]

Equation (2):

\[ 2x – y – 5 = 0 \Rightarrow y = 2x – 5 \]

Step 2: Prepare Tables of Values

For Equation (1): \(y = 5 – 3x\)

x	y
0	5
1	2

For Equation (2): \(y = 2x – 5\)

x	y
0	-5
3	1

Step 3: Graphical Representation

Plot the following points on the same Cartesian plane:

• Line 1: (0, 5) and (1, 2)
• Line 2: (0, −5) and (3, 1)

Join each pair of points to obtain two straight lines.

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

Result

The graphical solution of the given system of equations is:

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

Points Where the Lines Meet the Y-Axis

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

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

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

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

Conclusion

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

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

• (0, 5)
• (0, −5)