Problem 26 - Algebra and Trigonometry

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{aligned} 25 x-75 y &=100 \\\\-10 x+30 y &=-40 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Understanding the system

We are given a system of two linear equations: \(25x - 75y = 100\) and \(-10x + 30y = -40\). Our goal is to either find the point \((x, y)\) where the two lines intersect, or to show that the system has no solution (they are parallel), or infinitely many solutions (they are the same line).

Step 2: Simplifying the equations

Let's simplify both equations by dividing them by their greatest common factors. For the first equation, divide everything by 25: \(x - 3y = 4\). For the second equation, divide by 10: \(-x + 3y = 4\).

Step 3: Adding the simplified equations

Combine the equations \(x - 3y = 4\) and \(-x + 3y = 4\) by adding them together. This results in: \(x - 3y - x + 3y = 4 + 4\), which simplifies to \(0 = 8\).

Step 4: Analyze the result

The result from Step 3, \(0 = 8\), is a false statement. This indicates that the system of equations has no solution, meaning that the two lines are parallel and do not intersect at any point.

Key Concepts

linear equations

Linear equations are fundamental tools in algebra and analytical geometry. They represent lines in two-dimensional space and can typically be written in the form: \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Here are some essential points about linear equations:

• They graph as straight lines on the coordinate plane.
• Their solutions are points \((x, y)\) that satisfy both equations simultaneously.
• To solve a system of linear equations, we need to find where these lines intersect, if at all.

In the given exercise, the two linear equations represent lines that we need to analyze to determine their relationship in the coordinate plane.

no solution

In linear algebra, a system of equations may sometimes have no solution. This occurs when the equations represent two lines that never meet. A system like this is said to have 'no solution', meaning that it is inconsistent. In the given example, after simplifying the equation, we derived a contradiction: \(0 = 8\).

• This contradiction signals that there is no value of \(x\) and \(y\) that could make both simplified equations true at the same time.
• When such a false statement arises, the system has no consistent solutions.

Thus, in such cases, we can conclude that the lines don't intersect anywhere in the plane.

parallel lines

Parallel lines are two lines in a plane that never meet, no matter how far they are extended. They have the same slope, which ensures that the distance between them is constant, and they will never intersect. Here's how you can relate parallel lines to the problem we analyzed:

• For two lines to be parallel, their equations in slope-intercept form should show them having identical slopes.
• In our equation comparison \(x - 3y = 4\) and \(-x + 3y = 4\), both lines have equivalent opposite slopes, indicating they are parallel.
• Since they don't share any point of intersection, they will run parallel forever in one plane.

Recognizing parallel lines is crucial for quickly concluding that a system has no solution.

false statement

A false statement in a system of linear equations typically arises after manipulation or simplification. This commonly involves combining or eliminating variables but results in impossibility, such as \(0 = 8\). Here's how false statements help us understand the problem:

• They provide a clear indicator that a system of equations does not have a real solution.
• This invalid conclusion directly implies there is no common \((x, y)\) pair for the original lines, given they do not intersect.
• It further solidifies or leads to understanding other characteristics of the lines, such as being parallel in this case.

Recognizing and understanding false statements in linear systems quickly convey the inconsistency in parallel lines.