Problem 32 - Intermediate Algebra

Question

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{c} 2 x-3 y=4 \\ y=\frac{2}{3} x-\frac{4}{3} \end{array}\right) $$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \(y = \frac{2}{3}x - \frac{4}{3}\).

Step 1: Identify the Methods Available

We are given two equations: \(2x - 3y = 4\) and \(y = \frac{2}{3}x - \frac{4}{3}\). We can choose between substitution and elimination methods to solve the system. Since the second equation is already solved for \(y\), substitution is a more straightforward choice.

Step 2: Substitute the Expression for y

Substitute \(y = \frac{2}{3}x - \frac{4}{3}\) from the second equation directly into the first equation, obtaining: \(2x - 3\left(\frac{2}{3}x - \frac{4}{3}\right) = 4\).

Step 3: Simplify the Equation

Distribute \(-3\) in the equation: \(2x - 2x + 4 = 4\).

Step 4: Solve for x

Simplify the equation to find \(0 = 0\), which indicates that all terms cancel each other out, leading to the conclusion that the original system has infinitely many solutions. The solutions are given by the equation \(y = \frac{2}{3}x - \frac{4}{3}\).

Step 5: Interpret the Solution

Since \(0 = 0\) is a true statement, this implies that the two equations represent the same line. Therefore, every point on this line is a solution to the system.

Key Concepts

substitution method

The substitution method is a popular way to tackle systems of equations where one equation is already solved for one variable like our system here. It involves replacing a variable with its corresponding expression from another equation, thus reducing the number of variables. Since the second equation in the original system is already expressed as \( y = \frac{2}{3}x - \frac{4}{3} \), substitution is efficient in this scenario. Here's how it works:

• Start by taking the expression for \( y \) from the second equation.
• Substitute it into the first equation \( 2x - 3y = 4 \).
• The equation becomes \( 2x - 3(\frac{2}{3}x - \frac{4}{3}) = 4 \).

By substituting, we convert the system into a single-variable equation, simplifying the process and allowing us to find potential solutions. Remember, it's important to simplify correctly and keep the expressions neat for easy solving.

elimination method

The elimination method is another effective technique for solving systems of equations. Unlike substitution, it focuses on eliminating one of the variables by adding or subtracting equations to create an equation in one variable. This method can be useful when the coefficients of one variable are opposites or multiples across the equations. Consider this:

• Adjust or multiply equations so that adding or subtracting them cancels out one of the variables.
• Solve the resulting single-variable equation.
• Substitute back to find the other variable.

In our initial system \( 2x - 3y = 4 \) and \( y = \frac{2}{3}x - \frac{4}{3} \), elimination isn't the ideal choice, given that we'd first need to manipulate the second equation significantly. Thus, choosing substitution streamlines the process for this particular system.

infinitely many solutions

When dealing with a system of equations, encountering an identity like \( 0 = 0 \) signals a special case: infinitely many solutions. This means the equations describe the same line, and any point on this line is a valid solution. Let's put it into perspective:

• After simplifying, the equation reduces entirely, showing no distinct variable solutions.
• This indicates the original equations are not independent but rather, represent the same relationship.
• To express these solutions, you can use the equation of the line, such as \( y = \frac{2}{3}x - \frac{4}{3} \).

This characteristic is useful in understanding that, instead of a single-point solution, an entire set of coordinates satisfies the system. It's a fascinating result and a crucial insight into linear relationships.