Problem 22 - Algebra for College Students

Question

Solve each system by using the substitution method. \(\left(\begin{array}{l}y=5 x-9 \\ 5 x-y=9\end{array}\right)\)

Step-by-Step Solution

Answer

There are infinitely many solutions since the equations represent the same line.

Step 1: Substitute the Expression for y

We have the equations \( y = 5x - 9 \) and \( 5x - y = 9 \). Since \( y \) is already expressed in terms of \( x \) in the first equation, substitute \( y = 5x - 9 \) into the second equation.

Step 2: Solve for x

Substitute \( y = 5x - 9 \) into the second equation: \[ 5x - (5x - 9) = 9 \]Simplify the equation: \[ 5x - 5x + 9 = 9 \]\[ 9 = 9 \].Since this is an identity, the equation is true for all values of \( x \).

Step 3: Analyze the Result

The result \( 9 = 9 \) indicates that the two equations are essentially the same line, not distinct intersecting lines as initially presented. Therefore, they have infinite solutions.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations. It works by solving one equation for one variable and then substituting that expression into the other equation. This method is particularly useful when one of the equations is already solved for one of the variables. For example, in our given exercise, the equation is already solved for \( y \): \( y = 5x - 9 \).

Here’s how it works step by step:

• Identify an equation that is already solved for a variable. In this case, it's \( y = 5x - 9 \).
• Substitute this expression where the variable appears in the other equation. This replaces \( y \) in the second equation \( 5x - y = 9 \) with \( 5x - 9 \).
• Solve the resulting equation for the remaining variable.

The substitution method can simplify the process of solving systems, especially when dealing with linear equations.

Infinite Solutions

When solving a system of linear equations, you might encounter what's known as infinite solutions. This occurs when the two equations actually represent the same line. Thus, they intersect at every point along that line, leading to an infinite number of solutions.

In our problem, after substituting and simplifying, we end up with the equation \( 9 = 9 \). This is always true, indicating that every value of \( x \) satisfies the system.

Characteristics of systems with infinite solutions include:

• Both equations simplify to the same linear equation.
• The final result after substitution is a tautology, like \( 0 = 0 \) or \( 9 = 9 \).
• On a graph, both equations represent the same line.

Understanding infinite solutions is crucial in algebra as it broadens comprehension of how equations truly relate to one another.

Solving Linear Equations

Linear equations are equations between two variables that graph as straight lines. The solution to a system of linear equations can be a single point, no solutions, or infinite solutions.

Here’s a brief guide on solving linear equations within a system:

• Arrange both equations in standard form or solve one for one of the variables.
• Use substitution or elimination to find the values of the variables.
• Check if you have a unique solution, no solution, or infinite solutions by analyzing the final simplified equation.

In our example, by substituting and simplifying, we found that the solution results in \( 9 = 9 \), demonstrating that the system has infinite solutions.

Linear equations and their solutions are foundational in algebra, allowing us to tackle more complex equations and systems with confidence.