Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination. Label each line with its equation. $$\left\\{\begin{aligned} x-y &=2 \\ -2 x+2 y &=5 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system of equations has no solutions.

Step 1: Balance the Coefficients

Multiply the first equation by 2, which results in \(2x - 2y = 4\). So, now we have two equivalent equations with matching coefficients on x. Now they are: \(2x - 2y = 4\) and \(-2x + 2y = 5\).

Step 2: Eliminate x by Adding the Equations

Add the two equations \(2x - 2y + (-2x + 2y) = 4 + 5\). The left side simplifies to 0x and the right to 9, resulting in \(0x = 9\). Here, the solution seems impossible as 0 doesn't equal to 9. Hence, this system of equations has no solutions.

Step 3: Check the Results

As no x or y can satisfy both equations simultaneously, the system of equations is inconsistent and thus, there are no solutions.

Key Concepts

Understanding Systems of Equations

Systems of equations are sets of two or more equations where we are tasked with finding values for the variables that will satisfy all equations simultaneously. These are crucial when tackling problems with several dependent variables. Here we focus on a system of two linear equations:

• \( x - y = 2 \)
• \( -2x + 2y = 5 \)

These systems can be solved using various methods, such as substitution, graphing, or elimination. The choice of method often depends on the specific equations and personal preference. In our exercise, we've used the elimination method, which involves eliminating one variable by adding or subtracting equations after aligning their coefficients. This approach simplifies the system into a single-variable equation, making it easier to solve for remaining variables.

Identifying Inconsistent Systems

An inconsistent system of equations occurs when there are no possible solutions that satisfy all equations simultaneously. Essentially, this happens because the equations represent parallel lines that never intersect on a graph. Let's take a closer look at our example:

• We multiplied the first equation by 2, leading to \(2x - 2y = 4\).
• The second equation is \(-2x + 2y = 5\).

When we add these equations together, we reach a point where the variable terms are eliminated, resulting in \(0 = 9\). This is a clear contradiction because zero can never equal nine. Such a result indicates that the system is inconsistent as there's no overlap or common solution point between the lines in our graph.

Basics of Linear Equations

Linear equations are the cornerstone of high school algebra. These equations express relationships in a linear manner, forming straight lines when graphed. Each equation typically involves constants and variables, like \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
Solving linear equations is about finding the values of \(x\) and \(y\) that make the equation true. In systems of linear equations, as in our problem, each equation describes a line, and we are ultimately looking for a point (or points) where these lines intersect, revealing the solution(s).
However, as we've seen in this exercise, sometimes these lines don't intersect, leading to an inconsistent system. Whether consistent or not, understanding how to manipulate and interpret linear equations is essential for solving real-world problems using algebra.