Question
Solve the following system of equations graphically:
x + 2y = 8
2x − y = 6
Also, from the graph, find the vertices of the triangle formed by the two lines and the x-axis.
Solution — Step by Step
The graphical method asks us to plot two straight lines and find where they cross. Every linear equation gives a straight line — two points are enough to draw it.
Step 1: Find two points on the first line: x + 2y = 8
We find the intercepts — the easiest points to plot.
When x = 0: 0 + 2y = 8 → y = 4. Point: (0, 4)
When y = 0: x + 0 = 8 → x = 8. Point: (8, 0)
We have two points: (0, 4) and (8, 0). Draw a line through them.
Step 2: Find two points on the second line: 2x − y = 6
When x = 0: 0 − y = 6 → y = −6. Point: (0, −6)
When y = 0: 2x − 0 = 6 → x = 3. Point: (3, 0)
Two points: (0, −6) and (3, 0). Draw a line through them.
Step 3: Identify the intersection point from the graph.
The two lines intersect at (4, 2). This is the solution.
Step 4: Algebraic verification.
Line 1: 4 + 2(2) = 4 + 4 = 8 ✓ Line 2: 2(4) − 2 = 8 − 2 = 6 ✓
Solution
x = 4, y = 2 (intersection point)
Step 5: Find the triangle formed by the two lines and the x-axis.
The three vertices of the triangle are:
- A = (8, 0) — where line 1 meets the x-axis (x-intercept of line 1)
- B = (3, 0) — where line 2 meets the x-axis (x-intercept of line 2)
- C = (4, 2) — where the two lines intersect
These three points form a triangle with one side lying on the x-axis (from x = 3 to x = 8, length = 5) and the apex at (4, 2).
Area of the triangle (bonus): Base = AB = 8 − 3 = 5 units (along x-axis) Height = y-coordinate of C = 2 units
Area = (1/2) × base × height = (1/2) × 5 × 2 = 5 square units
Why This Works
Each linear equation in two variables graphs as a straight line. The solution to the system is the point(s) that satisfy BOTH equations simultaneously — geometrically, the point(s) where both lines exist at the same time, which is their intersection.
Two distinct non-parallel lines in a plane always intersect at exactly one point — giving one unique solution. Parallel lines never intersect — giving no solution. Coincident lines overlap completely — giving infinite solutions.
🎯 Exam Insider
In CBSE board exams, the graphical method question asks you to: (1) make a proper table of values, (2) draw neat axes with equal scales, (3) plot the points correctly, (4) draw both lines with a ruler, and (5) read and mark the intersection. Each of these steps carries marks. Messy graphs, unequal scales, or not using a ruler all cost presentation marks.
Alternative: Solving Algebraically
Adding the two equations: (x + 2y) + (2x − y) = 8 + 6 3x + y = 14 — hmm, that's not helpful directly.
Try elimination: multiply eq 1 by 1 and eq 2 by 2: x + 2y = 8 4x − 2y = 12
Add: 5x = 20 → x = 4, then y = 2. Confirms graphical answer.
Common Mistake
⚠️ Common Mistake
Mistake: Using only one point to draw each line.
One point is not enough to define a unique line — infinitely many lines pass through a single point. You need at least two points to draw a straight line. Always find and plot two points (preferably the x and y intercepts) for each equation. If you rush and plot only one point, your line will likely be wrong.