Problem 30 - Calculus: One and Several Variables

Question

Find the point where the lines intersect. $$l_{1}: 4 x-y=-2, \quad l_{2}: 3 x+2 y-0$$

Step-by-Step Solution

Answer

The two lines intersect at the point (-4/11, -6/11).

Step 1: Set Up Equations

The lines are given as \( l_{1}: 4x - y = -2 \) and \( l_{2}: 3x + 2y = 0 \). They intersect at a point \((x, y)\) that fulfills both equations.

Step 2: Solve the Equations System

We can solve this system of equations by the method of substitution or elimination. Here this solution will use elimination method.\nFirst, multiply \( l_{1} \) by 2 and \( l_{2} \) by 1 to make the coefficients of y in both equations equal, we get: \( l_{1}': 8x - 2y = -4 \) and \( l_{2}': 3x + 2y = 0 \).\nThen add these two equations together: \((8x - 2y) + (3x + 2y) = -4 + 0\), therefore \( 11x = -4 \). Divide both sides by 11 to solve for x, \( x = -4/11 \). \nPlug this x-value into the first original equation to solve for y: \( 4(-4/11) - y = -2 => y = (16/11) - 2 = -6/11 \)

Step 3: Find the Intersection Point

Substitute the solutions \( x = -4/11 \) and \( y = -6/11 \) back into point form to get the intersection point, which is \( (-4/11, -6/11) \).

Key Concepts

Intersection of Lines

When we talk about the intersection of lines, we're referring to a scenario where two lines meet at a single point on a graph. In mathematical terms, this is the point that satisfies the equations of both lines simultaneously.

Here, we have two lines denoted by their respective equations:

• Line 1: \(4x - y = -2\)
• Line 2: \(3x + 2y = 0\)

To find their intersection, we need to determine the values of \(x\) and \(y\) which fulfill both equations at the same time. This point is known as the solution to the system of equations, where the two lines overlap on the coordinate plane. Understanding this concept is crucial for deciphering how lines interact and form relationships in coordinate geometry.

Elimination Method

The elimination method is a structured approach for solving systems of linear equations. It's useful when dealing with equations that can be manipulated to cancel out one variable, allowing you to solve for the other. Let's see how it works in our example.

Given the equations:

• First Equation: \(4x - y = -2\)
• Second Equation: \(3x + 2y = 0\)

The goal is to eliminate one variable. Multiplying the first equation by 2 transforms it to have the same coefficient for \(y\) as the second equation (in absolute value). This gives us:

• Transformed First Equation: \(8x - 2y = -4\)
• Second Equation Stays: \(3x + 2y = 0\)

By adding these equations, the \(y\) terms cancel each other out, simplifying to \(11x = -4\). Now, simply solve for \(x\) by dividing both sides by 11, yielding \(x = -4/11\). Once \(x\) is found, we substitute back into one of the original equations to get \(y = -6/11\).

The elimination method makes it simpler to find solutions to otherwise complex systems by methodically removing one variable at a time.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, provides a bridge between algebraic equations and geometric structures on a graph, allowing us to visualize equations as lines and curves. It's all about understanding how to plot and interpret points, lines, and their interactions in the coordinate plane.

In our case, we are given the linear equations \(4x - y = -2\) and \(3x + 2y = 0\). These equations represent straight lines on a Cartesian plane. By solving them, as seen using the elimination method, the intersection point \((-4/11, -6/11)\) is found. This point is where the two lines cross paths, which can also be thought of as the graphical solution of the system.

The beauty of coordinate geometry lies in its ability to translate numerical relationships into geometric figures. It's a powerful tool for understanding and analyzing the visual representation of algebraic expressions, making problems like these not just abstract math exercises, but tangible and understandable challenges.