Problem 68 - Introductory Algebra for College Students

Question

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} \frac{x}{2}=\frac{y+8}{4} \\ \frac{x+3}{2}=\frac{y+5}{4} \end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations is inconsistent, indicating no solution exists.

Step 1: Clear Fractions by Multiplying by the LCD

First, notice that there are fractions with denominator 2 and 4 in both equations. Since 4 is the least common denominator (LCD) for both fractions, multiply each equation by 4 to remove fractions: \[4(\frac{x}{2})=4(\frac{y+8}{4})\] and \[4(\frac{x+3}{2})=4(\frac{y+5}{4}).\] This simplifies to \[2x=y+8\] and \[2(x+3)=y+5.\]

Step 2: Simplify the equations

Now, simplify the equations to a more standard form: \[2x=y+8\] becomes \[2x-y=8\] and \[2(x+3)=y+5\] simplifies to \[2x-y=-1.\]

Step 3: Solve the system using subtraction (elimination method)

Since both equations have \(2x-y\), this system of equations can be easily solved through subtraction to cancel out \(y\). The first step would be to subtract the second equation from the first, resulting in \[2x - y - (2x - y) = 8 - (-1)\] which simplifies to \[0 = 9.\]

Step 4: Conclusion of the solution

Since the equation obtained (0=9) is not a true statement, it indicates that the original pair of equations has no solution. Hence, the system is inconsistent.

Key Concepts

Linear Equations

A linear equation is a mathematical statement that describes a line through the points in a plane, often written in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. Linear equations can have one or more variables, and they exclusively contain terms that are either constants or coefficients multiplied by a variable.
In systems of linear equations, such as the one in the exercise, you are dealing with two or more linear equations simultaneously. The goal is to find a common solution for all equations, which occurs at the intersection point of the lines they represent. Sometimes, systems can be inconsistent, meaning no common solution exists. In other cases, they may be dependent, implying infinitely many solutions.
The initial set of equations given involves fractions:

• \( \frac{x}{2} = \frac{y+8}{4} \)
• \( \frac{x+3}{2} = \frac{y+5}{4} \)

By simplifying these equations, we remove the fractions and reveal the linear forms \( 2x - y = 8 \) and \( 2x - y = -1 \). This reveals the linear systems nature of the problem.

Substitution Method

The substitution method is a way to solve systems of linear equations by solving one equation for one variable and then substituting this expression into another equation. This helps to reduce the system to a single equation in one variable.
Although the problem in the exercise was approached using the elimination method, understanding substitution is very helpful. Here’s how you would typically apply it:

• First, isolate one variable in one of the equations. Suppose you rearrange \( 2x = y + 8 \) to \( y = 2x - 8 \).
• Next, substitute this expression for \( y \) in the other equation: \( 2x - (2x - 8) = -1 \).
• By substituting, if the variable’s value makes the system impossible (like \( 0 = 9 \)), you identify the system as inconsistent.

While substitution wasn’t the method used in the exercise, it can always be an alternative approach, especially when handling simpler systems or when one equation is already solved for a variable.

Elimination Method

The elimination method involves using addition or subtraction to eliminate one variable, making it easier to solve the resulting single-variable equation. This can be particularly effective when equations are aligned or have coefficients that are easily manipulated.
In the given system:

• \( 2x - y = 8 \)
• \( 2x - y = -1 \)

Both equations have the same left-hand side, \( 2x - y \). When you subtract the second equation from the first, you attempt to eliminate both \( x \) and \( y \):
\[ (2x - y) - (2x - y) = 8 - (-1) \]
Double subtraction leads to \( 0 = 9 \), an impossibility. This contradiction confirms that no solutions exist for the system, classifying it as inconsistent.
The elimination method here elegantly reveals the nature of the system, showing quickly and clearly that the equations don’t intersect at any point on a graph, highlighting the absence of a solution.