Problem 39 - 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. \(\frac{x}{4}-\frac{y}{4}=-1\) \(x+4 y=-9\)

Step-by-Step Solution

Answer

The solution to the system of equations is x=-3 and y=1.

Step 1: Rearrange the First Equation for x

Rewrite the first equation \(\frac{x}{4}-\frac{y}{4}=-1\) to solve for x. You could multiply every term by 4 to eliminate the fractions. Then x=y-4.

Step 2: Substitute x into the Second Equation

After we found that x=y-4, replace x in the second equation \(x+4y=-9\). This results to: (y-4)+4y=-9. Simplify to 5y-4=-9

Step 3: Solve for y

Solving the aforementioned equation for y gives us y=1.

Step 4: Substitute y back into the rearranged First Equation

Substitute y=1 into the rearranged first equation from step 1 (x=y-4). We find x=-3.

Step 5: Verification

Substitute x=-3 and y=1 into both original equations to verify that these values are correct and satisfy both equations.

Key Concepts

Set Notation

Set notation is a key concept in mathematics used to express complex sets clearly and concisely. When solving systems of equations, like the one in the exercise, set notation helps to define possible values for variables that satisfy the given equations. In the context of linear equations, solutions can be:

• A single point, noted as \((x, y)\)\.
• Infinitely many solutions, indicating an overlap of all solutions in the equations.
• No solution, when the equations never intersect.

In the exercise, once you find the solution(s) to the system, you use set notation to present it. For example, if there's one solution, it is written as \((x, y)\)\. If there are infinitely many solutions, it is described in terms of one variable, like \("(x, y) = (t, 3 - 2t)"\)\. No solution is expressed by an empty set, denoted as \("\emptyset\"\)\.
Understanding set notation allows you to communicate the solution sets effectively.

Infinitely Many Solutions

A system of equations has infinitely many solutions when all the equations describe the same line, meaning they overlap completely. This happens when each equation is a multiple or transformation of the other equation, resulting in an overlapping solution.
To determine if a system has infinitely many solutions:

• Rearrange and simplify the equations to see if they are not independent.
• Both equations will simplify to form one identity, such as \(0 = 0\), thus covering all possible points on the line.

Representing this situation using variables, one variable can be expressed in terms of another, setting a parametric equation like \( \(x = 2t, y = 3 - 2t \) \)\ for all real numbers \( t \). Such solutions often reflect an infinite nature of overlap between the equations.
In cases with infinitely many solutions, the answer is often given using a set notation to express the dependent relationship between the variables.

No Solution

A system of equations may have no solution if the lines represented by the equations never intersect, which generally means they are parallel. This situation can arise due to:

• Different constant terms in otherwise proportional equations.
• Aligning slopes but differing intercepts.

In practice, you can recognize no solution situations by simplifying both equations and checking if they end with something contradictory, like \(0 = 5\). Such an equation tells that there's no \(x, y\) pair satisfying both equations simultaneously.
In set notation, a system with no solutions is expressed as an empty set, \(\emptyset\), indicating there's no possible solution for the system. Understanding these conditions equips you to recognize systems with no solutions and articulate them in mathematical terms.

Substitution Method

The substitution method is a powerful technique for solving systems of equations whereby you express one variable in terms of the others and substitute this back into one of the original equations. Let's break it down:

• Solve for one variable within one equation.
• Substitute this expression into the other equation.

In the given exercise, we first rearranged \(\frac{x}{4} - \frac{y}{4} = -1\) to isolate \(x\), obtaining \(x = y - 4\). Then, substituting \(x = y - 4\) into the second equation \(x + 4y = -9\), simplifies the system to a single-variable equation. Solving this gives a specific value, say for \(y\), which can then be back-substituted to find the value of \(x\).
This method is very effective for systems where it is easy to isolate one variable and makes solving complex systems more manageable. Mastering substitution aids in understanding critical systems of linear equations by breaking them into simpler, two-step processes.