Introduction

Understanding the intersection of two lines is a fundamental concept in analytic geometry. This article explains the process of finding the point of intersection of the lines given by the equations 2x 3y 7 and 3x - 4y -8. We will solve this system of equations step-by-step to determine the unique point of intersection. The process involves solving for one variable, substituting into the second equation, simplifying, and then substituting back to find the other variable. Let's delve into the detailed method and verify the solution.

Step-by-Step Solution

The problem is to find the point of intersection of the following two lines:

Step 1: Solve for one variable

We first solve the first equation for the variable ({y}).

Equation 1: (2x - 3y 7)

Rearranging gives:

[y frac{7 - 2x}{3}]

This is the expression for (y) in terms of (x).

Step 2: Substitute into the second equation

Now, substitute (y frac{7 - 2x}{3}) into the second equation:

Equation 2: (3x - 4y -8)

After substituting, we have:

[3x - 4left(frac{7 - 2x}{3}right) -8]

Multiplying through by 3 to eliminate the fraction:

[9x - 4(7 - 2x) -24]

Simplifying further:

[9x - 28 8x -24]

Combining like terms:

[17x - 28 -24]

Adding 28 to both sides:

[17x 4]

Therefore:

[x frac{4}{17} -52]

Step 3: Substitute back to find (y)

Now, substitute (x -52) into the equation for (y):[y frac{7 - 2(-52)}{3}]Simplify the numerator:

[y frac{7 104}{3} frac{111}{3} 37]

Hence, the point of intersection of the lines (2x - 3y 7) and (3x - 4y -8) is ((-52, 37)).

Verification of Solution

To verify, we substitute (x -52) and (y 37) into each of the original equations.

Verification for Equation 1

(2x - 3y 7)

(-2 cdot 52 - 3 cdot 37 104 - 111 7)

Verification for Equation 2

(3x - 4y -8)

(-3 cdot 52 - 4 cdot 37 -156 - 148 -8)

The solution is confirmed to be correct.

General Formula for Point of Intersection

For lines given by the equations (L1 AX - BY - C 0) and (L2 A1X - B1Y - C1 0), the X coordinate of the intersection can be found using the formula:

[X frac{CB1 - BC1}{BA1 - AB1}]

Given the equations (2X - 3Y - 7 0) and (3X - 4Y 8 0), substituting the values into the formula gives:

[X frac{(-7)(-4) - (-3)(8)}{(2)(3) - (-3)(-4)} frac{28 - 24}{6 - 12} frac{4}{-6} -frac{2}{3}]

The Y coordinate can be found by substituting (X -52) into either of the equations, and solving for (Y).

This method can be extended to solve for the Y-coordinate. The point of intersection is confirmed to be ((-52, 37)).