Problem 41 - College Algebra Essentials

Question

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. $$ \left\\{\begin{array}{l} 2 x=3 y+4 \\ 4 x=3-5 y \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \{(\frac{3}{2}, -\frac{5}{11})\}

Step 1 - Isolate Variable in First Equation

Isolate one variable in the first equation. Let's isolate \(x\) in terms of \(y\). This gives us: \(x = \frac{3y+4}{2}\)

Step 2 - Substitute in Second Equation

Substitute \(x\) from the first equation into the second equation: \(4\left(\frac{3y+4}{2}\right) = 3-5y\)

Step 3 - Simplify and Solve for y

Simplify the equation and solve for \(y\): After simplification, we get\(6y + 8 = 3 - 5y\), that further simplifies into \(11y = -5\), and finally we get \(y = -\frac{5}{11}\)

Step 4 - Substitute y into First Equation

Substitute this value of \(y\) into the isolated x equation: \(x = \frac{3(-\frac{5}{11})+4}{2}\)

Step 5 - Simplify and Solve for x

Simplify the above equation to solve for \(x\). After simplification, we get \(x = \frac{33}{22}\) or \(x = \frac{3}{2}\)

Step 6 - Expressing Solution in Set Notation

Finally, the solution to the exercise can be expressed in set notation as: \{(\frac{3}{2}, -\frac{5}{11})\}

Key Concepts

Solution Set

The concept of a solution set is central to understanding systems of equations. It refers to the complete collection of solutions that satisfy all the equations in a system. In a system of linear equations like the one we're dealing with, each solution is a pair of values for the variables that make each equation true simultaneously. Consider this system:

• \(2x = 3y + 4\)
• \(4x = 3 - 5y\)

The solution set for this system is derived from finding the specific values of \(x\) and \(y\) that satisfy both equations. Once these values are determined, they can be represented in a set notation, often looking like this: \(\{(x, y)\}\). For this exercise, the solution set was found to be \(\{(\frac{3}{2}, -\frac{5}{11})\}\). This indicates that \(x = \frac{3}{2}\) and \(y = -\frac{5}{11}\) make both equations valid.

Solving Methods

There are several methods available for solving systems of linear equations, each with its own strengths. The two most common are substitution and elimination.

• Substitution Method: This involves solving one of the equations for one variable and then substituting that expression into the other equation.
• Elimination Method: This method eliminates one variable by adding or subtracting equations after making their coefficients equal.

Each method can be more or less efficient depending on the system at hand. In our exercise, we opted for the substitution method because it makes isolating a single variable straightforward. This allows us to efficiently determine the solution set.

Substitution Method

The substitution method involves a series of logical steps to isolate and substitute variables, making it a systematic approach to solving linear equations. Here’s how it was applied in our system:

First, choose one equation and solve for one variable. In our case, we isolated \(x\) in the first equation:

• \(x = \frac{3y+4}{2}\)

Second, substitute the expression for \(x\) into the second equation:

• \(4\left(\frac{3y+4}{2}\right) = 3 - 5y\)

This substitution transforms the problem into a single-variable equation, facilitating easier manipulation. Solve this equation to find \(y\), and then substitute \(y\)'s value back into the expression for \(x\). This process unravels the values of both variables in steps and helps to clearly identify the solution set.

Infinitely Many Solutions

A system of linear equations may not always have a unique solution like in our exercise. Sometimes, a system can have infinitely many solutions. This occurs when the equations are dependent, or in other words, they represent the same line in a two-dimensional space.

• For example, if simplifying one equation results in an identical equation as the other, they are essentially expressing the same constraint.
• Graphically, this means the two lines coincide, which visually translates to infinite intersection points.

When a system exhibits infinitely many solutions, the solution set can be expressed as a range or parametric form reflecting all possible solutions. These types of systems are crucial to recognize as they indicate a different relationship between variables, and often imply redundancy in constraints.