Problem 35 - College Algebra

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. \(3 x-2 y=-5\) \(4 x+y=8\)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. \(3 x-2 y=-5\) \(4 x+y=8\)

Step-by-Step Solution

Answer

The solution to this system of equations is \(\{(3/7, 44/7)\}\).

Step 1: Organize the System on Paper

Rewrite the system so it is clear which equation corresponds to which variable. \[ \begin{{align*}} 3x - 2y &= -5 \ (1) \ 4x + y &= 8 \ (2) \end{{align*}} \]

Step 2: Implement the Elimination Method

To eliminate \(y\), multiply the second equation by 2 and add it to the first equation. This gives a new system of equations \[ \begin{{align*}} 3x - 2y + 4x + 2y & = -5 + 8 \ -------- 7x & = 3 \end{{align*}} \] The solution to this equation is \(x = 3/7.\)

Step 3: Solve for the Remaining Variable

Now substitute \(x = 3/7\) into the second equation of the original system to solve for \(y: 4(3/7) + y = 8\), which simplifies to \(12/7 + y = 8\), and then \(y = 8 - 12/7 = 56/7 - 12/7 = 44/7.\)

Step 4: Express the Solution as a Set

The solution to this system of equations is the set of all \((x, y)\) such that \(x = 3/7\) and \(y = 44/7\). This can be expressed in set notation as \(\{(3/7, 44/7)\}.\)

Key Concepts

Elimination Method

In algebra, solving systems of equations can often be achieved using various methods. One effective approach is the elimination method, which simplifies the equations to make solving easier. In the elimination method, the goal is to eliminate one variable by making its coefficients in the equations cancel out. This is done by adding or subtracting the equations after they have been appropriately modified.

Here’s how it works for the given example:

• First, identify the equations in the system. For instance, with the equations \(3x - 2y = -5\) and \(4x + y = 8\), you can see that the coefficients of \(y\) are \(-2\) and \(1\) respectively.
• To eliminate \(y\), we multiply the second equation by \(2\) to get \(8x + 2y = 16\).
• Now, add this new equation to the first one, \(3x - 2y = -5\), which results in eliminating \(y\) and simplifies to \(7x = 3\).

This process reduces the system to a single equation in one variable, making it easier to find one of the original variables.

Once that is achieved, you can substitute back to find the other variable, completing the method.

Set Notation

In mathematics, set notation is a standard way to denote the solution sets of equations or systems of equations. It helps communicate results clearly and concisely. When talking about set notation in solving systems of equations, we refer to representing the solution as ordered pairs—in this case, a pair \((x, y)\).

For example, after solving a system of equations, you may express the solution in set notation as \(\{(x, y)\}\). In the exercise provided, after using the elimination method, the values found were \(x = \frac{3}{7}\) and \(y = \frac{44}{7}\). Thus, the solution set is written in set notation as \(\{( \frac{3}{7}, \frac{44}{7} )\}\).

This notation effectively communicates that there is only one unique solution to the system.

Solution Sets

Understanding solution sets is crucial in solving systems of equations. Solution sets tell us the collection of all possible solutions that satisfy the given equations. In the realm of algebra, a system of equations can have:

• One unique solution
• No solution
• Infinitely many solutions

For the given exercise, the elimination method yielded a single solution \((x, y)\) as \(\left( \frac{3}{7}, \frac{44}{7} \right)\). This indicates that the solution set consists of this one unique point. It's important to highlight that different scenarios can arise, such as parallel lines resulting in no solution or coincident lines leading to infinitely many solutions, described with set notation like \(\emptyset\) or an expression containing parameters.

Algebraic Manipulation

Algebraic manipulation is a key tool when solving equations, especially systems of equations. It involves rearranging and simplifying equations to identify solutions or make the process more manageable. This can include:

• Adding or subtracting equations to eliminate variables (as seen in the elimination method).
• Multiplying entire equations by constants to align coefficients.
• Substituting values to solve for unknowns after reducing the system.

In our example, after using the elimination method to find \(x = \frac{3}{7}\), it was necessary to use algebraic manipulation again.

This was done by substituting \(x = \frac{3}{7}\) back into one of the original equations, specifically \(4x + y = 8\), to find \(y = \frac{44}{7}\).

Such manipulation ensures that all steps maintain the balance and equality of the original equations, leading to the correct solution.