Draw the graphs of the equations 3x – 2y = 4 and x + y – 3 = 0. On the same graph paper,
The graphs of 3x – 2y = 4 and x + y – 3 = 0 intersect at the point (2, 1), obtained by plotting the given coordinates and drawing both lines on the same graph paper.
Draw the graphs of the equations 3x – 2y = 4 and x + y – 3 = 0. On the same graph paper, find the coordinates of the point were the two graph lines intersect.
$3 x-2 y=4$
$\Rightarrow 2 y=3 x-4$
$\Rightarrow y=\frac{3 x-4}{2}$
When $x=0, y=\frac{3 \times 0-4}{2}=\frac{0-4}{2}=\frac{-4}{2}=-2$
When $x=2, y=\frac{3 \times 2-4}{2}=\frac{6-4}{2}=\frac{2}{2}=1$
When $x=-2, y=\frac{3 \times(-2)-4}{2}=\frac{-6-4}{2}=\frac{-10}{2}=-5$
Thus, the points on the line 3x – 2y = 4 are as given in the following table:
Plotting the points (0, –2), (2, 1) and (–2, –5) and drawing a line passing through these points, we obtain the graph of of the line 3x – 2y = 4.
$x+y-3=0$
$\Rightarrow y=-x+3$
When $x=0, y=-0+3=3$
When $x=1, y=-1+3=2$
When $x=-1, y=-(-1)+3=1+3=4$
Thus, the points on the line x + y – 3 = 0 are as given in the following table:
Plotting the points (0, 3), (1, 2) and (–1, 4) and drawing a line passing through these points, we obtain the graph of of the line x + y – 3 = 0.
It can be seen that the lines 3x – 2y = 4 and x + y – 3 = 0 intersect at the point (2, 1).
Frequently Asked Questions
Find answers to common questions.
What is the intersection point of 3x – 2y = 4 and x + y – 3 = 0?
The two lines intersect at the point (2, 1). This means x = 2 and y = 1 is the only pair of values that satisfies both equations simultaneously. You can verify this by substituting into each equation: 3(2) – 2(1) = 4 ✓ and 2 + 1 – 3 = 0 ✓.
How do I choose values of x when making a table for graph plotting?
Choose x values that are easy to compute and give whole-number or simple fractional y values. For 3x – 2y = 4, using x = 0, 2, and –2 gives y = –2, 1, and –5 — all integers. Avoid x values that produce large decimals, as these are harder to plot accurately on graph paper.
Why do we need at least three points to draw a line on a graph?
Strictly speaking, two points define a unique straight line. However, a third point acts as a verification check. If your third point does not lie on the line joining the first two, you have made an arithmetic mistake somewhere. This habit of using three points prevents errors before they cost marks.
How can I predict which x values to substitute into the equation?
Start with x = 0 (gives the y-intercept directly) and x values that cancel out fractions in the formula. For y = (3x – 4)/2, choosing even values of x ensures the numerator is even and divides cleanly. Try x = 0, 2, 4 or x = 0, 2, –2. If the equation has no fractions, any simple integers work.
Can two linear equations have no intersection point? When does that happen?
Yes. If two linear equations represent parallel lines, they never intersect and there is no solution — called an inconsistent pair. Parallel lines have equal slopes but different y-intercepts. For example, 2x + 3y = 6 and 2x + 3y = 12 are parallel. If two lines coincide (same slope and same intercept), there are infinitely many solutions.