Solve Graphically the System of Linear Equations and Find the Points Where the Lines Meet the Y-Axis: 2x − 5y + 4 = 0, 2x + y − 8 = 0

Video Explanation

Watch the video below to understand the complete solution step by step:

Solution

Question:
Solve graphically the following system of linear equations. Also find the coordinates of the points where the lines meet the y-axis:
2x − 5y + 4 = 0
2x + y − 8 = 0

Step 1: Rewrite the Equations in Slope-Intercept Form

For 2x − 5y + 4 = 0:

−5y = −2x − 4
y = (2/5)x + 4/5

For 2x + y − 8 = 0:

y = 8 − 2x

Step 2: Find the Points Where the Lines Meet the Y-Axis

A line meets the y-axis where x = 0.

For 2x − 5y + 4 = 0:

Putting x = 0:
−5y + 4 = 0 ⇒ y = 4/5
So, the line meets the y-axis at (0, 4/5).

For 2x + y − 8 = 0:

Putting x = 0:
y = 8
So, the line meets the y-axis at (0, 8).

Step 3: Graphical Interpretation

When the graphs of both equations are drawn on the same Cartesian plane, the two straight lines intersect at exactly one point.

The point of intersection of the two lines represents the graphical solution of the given system.

Step 4: Find the Point of Intersection

Solving the equations simultaneously:

2x − 5y + 4 = 0
2x + y − 8 = 0

Subtracting the first equation from the second:

6y − 12 = 0 ⇒ y = 2

Substituting y = 2 in 2x + y − 8 = 0:

2x + 2 − 8 = 0 ⇒ 2x = 6 ⇒ x = 3

Final Answer

∴ The graphical solution of the given system of equations is (3, 2).

The points where the lines meet the y-axis are (0, 4/5) and (0, 8).

Conclusion

Since the two straight lines intersect at one point, the system of linear equations has a unique solution. The lines meet the y-axis at (0, 4/5) and (0, 8) respectively.