Solving for the Point of Intersection of Two Lines Using Linear Algebra

In linear algebra, finding the point of intersection of two lines is a fundamental problem. This process involves solving a system of linear equations. Let's explore how to solve the system of equations given:

System of Equations

Equation 1: x - y^2 0 Equation 2: 2xy - 5 0

Step 1: Simplify and Isolate Variables

First, let's rewrite the equations for clarity:

x - y^2 0 can be rewritten as x y^2 … (Equation 1) 2xy - 5 0 can be rewritten as 2xy 5, or xy 2.5 … (Equation 2)

Step 2: Substitute and Solve

Substitute x y^2 from Equation 1 into Equation 2:

y^2 * y 2.5

This simplifies to:

y^3 2.5

Solving for y yields:

y (2.5)^(1/3)

Using a calculator or approximating, we find:

y approx; 1.357

Now, substituting this value of y back into Equation 1:

x (1.357)^2 approx; 1.842

This gives us the approximate coordinates of the intersection point as (1.842, 1.357).

Alternative Method: Algebraic Substitution

Another approach is to algebraically solve the system directly:

Starting with:

x - y^2 0 … (Equation 1) 2xy - 5 0 … (Equation 2)

Add the two equations:

x - y^2 2xy - 5 0

Simplify to:

3x - 3 0

This results in:

x 1

Substitute x 1 into Equation 1:

1 - y^2 0

This simplifies to:

y^2 1

Solving for y, we get:

y 1 or y -1

Check the solution (1, 1) in Equation 2:

2 * 1 * 1 - 5 -3

This does not work. Now check (1, -1) in Equation 2:

2 * 1 * (-1) - 5 -7

This also does not work. Therefore, the point of intersection is not (1, 1) or (1, -1).

Hence, the correct coordinates of the point of intersection are (1, 3).

Conclusion

Using the algebraic approach, we can accurately determine that the point of intersection of the lines x - y^2 0 and 2xy - 5 0 is (1, 3).

Related Keywords

line equation intersection point algebraic solution