Problem 25 - College Algebra

Question

Solve each system by addition. $$ \begin{array}{l} -x+2 y=-1 \\ 5 x-10 y=6 \end{array} $$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Identify the Equations

Start by clearly identifying the given system of equations: 1. \(-x + 2y = -1\)2. \(5x - 10y = 6\).

Step 2: Align Coefficients for Elimination

Notice that the coefficient of \(y\) in both equations can facilitate elimination. The coefficients \(2y\) and \(-10y\) can be aligned by multiplying the first equation by 5, which gives:\[-5x + 10y = -5\].

Step 3: Add the Equations

Now, add the aligned equations to eliminate \(y\):\[(-5x + 10y) + (5x - 10y) = -5 + 6\]This simplifies to:\[0 = 1\].

Step 4: Evaluate the Result

The equation \(0 = 1\) is a contradiction, meaning it is not possible for both equations to be satisfied simultaneously. This indicates that the system has no solution.

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of variables. These systems can be used to find values for the variables that satisfy all the equations simultaneously. In the example given, we have two equations involving two variables, \(x\) and \(y\):

• \(-x + 2y = -1\)
• \(5x - 10y = 6\)

The goal is to find values for \(x\) and \(y\) that make both equations true. There are various methods to solve such systems, including substitution, graphing, and elimination. The elimination method is often used when the coefficients of one variable can be easily manipulated to cancel out one another.

Elimination Method

The Elimination Method is a reliable technique for solving systems of linear equations. It involves adding or subtracting the equations to eliminate one of the variables. This process allows us to focus on solving for one variable first. To use elimination effectively, follow these steps:

• Align Coefficients: Make the coefficients of one variable the same in both equations. In this exercise, the coefficients of \(y\) were manipulated to both be 10 by multiplying the first equation by 5.
• Add or Subtract Equations: Combine the equations to eliminate the chosen variable. For example, adding our new equation \(-5x + 10y = -5\) and \(5x - 10y = 6\) will help eliminate \(y\).
• Solve for Remaining Variable: You'll be left with an equation that involves only one variable, which can be solved directly.

By using the elimination method, we aim to simplify the process of finding solutions that satisfy the entire system.

No Solution Systems

Sometimes, a system of equations may not have a solution. This happens when the attempts to eliminate variables result in a contradiction. In our exercise, after applying the elimination method, we ended up with the expression \(0 = 1\). This is a classic sign that the system is inconsistent.

• Understanding Contradictions: If simplifying the system of equations leads to a statement that is always false, like \(0 = 1\), it indicates there is no set of two numbers that can satisfy both equations simultaneously.
• Geometrical Interpretation: Geometrically, this means that the lines represented by the equations are parallel and will never intersect.
• Identifying No Solution Systems: While solving, recognize these results as no solution scenarios, meaning the original equations do not work together to produce an answer.

It is important to recognize that not all systems of equations can be solved, but understanding why provides greater insight into the relationships between the equations themselves.