Problem 51 - Intermediate Algebra

Question

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} \frac{x}{4}=1+\frac{y}{5} \\ x=\frac{4}{5}(y+10) \end{array}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent with no solutions.

Step 1: Simplify the First Equation

The first equation is \( \frac{x}{4} = 1 + \frac{y}{5} \). Multiply every term by 20 to eliminate the fractions: \( 20 \cdot \frac{x}{4} = 20 \cdot 1 + 20 \cdot \frac{y}{5} \). This simplifies to \( 5x = 20 + 4y \). Rearrange to form: \( 5x - 4y = 20 \).

Step 2: Simplify the Second Equation

The second equation is \( x = \frac{4}{5}(y + 10) \). Distribute \( \frac{4}{5} \) into the expression: \( x = \frac{4}{5}y + 8 \). Multiply every term by 5 to eliminate the fraction: \( 5x = 4y + 40 \).

Step 3: Set Up the System of Equations

After simplification, the system of equations is: \( \begin{cases} 5x - 4y = 20 \ 5x = 4y + 40 \end{cases} \). The second equation can be rearranged to match the form of the first: \( 5x - 4y = 40 \).

Step 4: Compare and Solve Equations

Compare \( 5x - 4y = 20 \) and \( 5x - 4y = 40 \). These equations are parallel lines with different intercepts, meaning they will never intersect. Thus, the system has no solutions.

Key Concepts

Linear Equations

Linear equations form the foundation of algebraic systems in mathematics. They are equations of the first degree, meaning the variables appear with an exponent of one. In general, a linear equation in two variables can be represented as \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
A characteristic feature of linear equations is that their graph forms a straight line. For instance, in our exercise, both simplified equations, \( 5x - 4y = 20 \) and \( 5x = 4y + 40 \), are linear equations. After rearranging, the second equation also becomes \( 5x - 4y = 40 \).
Here are a few properties of linear equations:

• They can be plotted on a graph as straight lines.
• The solution to a linear equation is usually a point on this line.
• For a system of linear equations, the solution is the point where the equations intersect.

Understanding how to manipulate these equations is crucial, as it allows us to analyze systems and find solutions or determine characteristics like dependence or inconsistency.

Inconsistent Systems

An inconsistent system of equations is a system that has no solution. This means that the equations represent parallel lines that never intersect. In the context of our exercise, after simplifying the original equations, we discovered they correspond to the following simplified system:

• \( 5x - 4y = 20 \)
• \( 5x - 4y = 40 \)

The above equations both have the same left-hand side but different constants on the right-hand side. This form, \( 5x - 4y = c \), where \( c \) is different for both equations, indicates that the lines are parallel. They have the same slope but different y-intercepts.
Here's how you can identify an inconsistent system:

• Equations that have the same coefficients for the variables but different constants.
• On a graph, you will see two lines that never meet.

The key takeaway is that no value of \( x \) and \( y \) will ever satisfy both equations simultaneously.

Dependent Equations

Dependent equations, unlike inconsistent systems, are equations that have infinitely many solutions. This occurs when the equations essentially describe the same line. When working with systems of equations, dependent equations will have the same slope and the same y-intercept.
Although dependent equations don't occur in our current exercise, it's useful to understand this concept. In reality, if you were to simplify two different equations in a system and end up with the exact same equation, like \( 5x - 4y = 20 \) and \( 5x - 4y = 20 \), then the system is dependent.
Here are the hallmarks of dependent equations:

• They produce overlapping or coincident lines on a graph.
• The system shares the same slope and intercept, meaning every solution that fits one fits both.

In essence, a dependent system isn't truly a set of different equations; instead, it's a repetition, highlighting the infinite solutions along a single line. Recognizing this in real-world problems can save you much time by avoiding unnecessary calculations.