Question

Solve the following system of equations graphically:

2x – 3y = 1, 4x – 3y + 1 = 0

Answer 1

1. Express equations in terms of y

To plot the lines easily, rewrite both equations in slope-intercept form (y = mx + c):

Equation 1:

2x – 3y = 1

⇒ 3y = 2x – 1

⇒ `y = (2x - 1)/3`

Equation 2:

4x – 3y + 1 = 0

⇒ 3y = 4x + 1

⇒ `y = (4x + 1)/3`

2. Find coordinate points for both lines

Find a few coordinate pairs (x, y) that satisfy each equation to accurately draw the straight lines on a Cartesian plane.

For Line 1 (2x – 3y = 1):

If y = 1 ⇒ 2x – 3(1) = 1

⇒ 2x = 4

⇒ x = 2

Point: (2, 1)

If x = –1 ⇒ 2(–1) – 3y = 1

⇒ –3y = 3

⇒ y = –1

Point: (–1, –1)

If x = 5 ⇒ 2(5) – 3y = 1

⇒ –3y = –9

⇒ y = 3

Point: (5, 3)

For Line 2 (4x – 3y + 1 = 0):

If y = 1 ⇒ 4x – 3(1) + 1 = 0

⇒ 4x = 2

⇒ `x = 1/2`

Point: (0.5, 1)

If x = –1 ⇒ 4(–1) – 3y + 1 = 0

⇒ –3y = 3

⇒ y = –1

Point: (–1, –1)

If x = 2 ⇒ 4(2) – 3y + 1 = 0

⇒ –3y + 1 = 0

⇒ –3y = –9

⇒ y = 3

Point: (2, 3)

3. Plot the system of equations

Plotting the calculated coordinates and drawing a line through each set yields the following visual behavior:

4. Locate the intersection point

By looking at the graph, both lines intersect precisely at one common coordinate location:

P = (–1, –1)

Since the point of intersection satisfies both equations simultaneously, it represents the real solution to the system.