Problem 104 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} y=-\frac{1}{4} x \\ x+4 y=8 \end{array}\right. $$

Step-by-Step Solution

Answer

No solution; the system is inconsistent.

Step 1: Substitute y in the second equation

Insert the expression for y from the first equation into the second equation. Given y = -\( \frac{1}{4} \), substitute it into the second equation: \[ x + 4 \left( -\frac{1}{4} x \right) = 8 \].

Step 2: Simplify the equation

Simplify the expression within the equation from Step 1:\[ x + 4 \left( -\frac{1}{4} x \right) = 8 \] simplifies to \[ x - x = 8 \].

Step 3: Solve for x

Combine like terms in the equation: \[ x - x = 8 \] simplifies to \[ 0 = 8 \].

Step 4: Interpret the result

Since 0 does not equal 8, there is no solution. The system of equations is inconsistent.

Key Concepts

Substitution Method

The substitution method is used to solve systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation.

In the given problem, we are provided with the equations:
\[\begin{equation}\begin{cases}y = -\frac{1}{4}x \x + 4y = 8\end{cases}\end{equation}\]
The goal is to replace one variable to simplify the process.
Steps:

• Solve one equation for one variable.
• Substitute that expression into the other equation.
• Simplify and solve the resulting equation.
• Use the solution to find the other variable.

In our exercise, we solved for y from the first equation:
y = -\( \frac{1}{4}x \).
We then substituted y into the second equation and simplified.
This method helps pinpoint possible solutions or inconsistencies in the system.

Inconsistent Systems

An inconsistent system of equations means that the equations do not have any solutions that satisfy both equations simultaneously.

When we simplified the system, we got:
x - x = 8
which simplified to:
0 = 8.
This equation is false. Hence, there's no solution for this system.

Properties of inconsistent systems:

• They have no points of intersection if graphed.
• The equations represent parallel lines.
• The lines have the same slope but different intercepts.

Inconsistent systems can arise in various real-world situations, such as constraints that cannot all be satisfied at once.

Solving Algebraic Equations

Solving algebraic equations involves manipulating the equations to find the values of the unknown variables.

We use various methods like substitution, elimination, and graphical methods.

When simplifying equations:

• Combine like terms
• Isolate variables on one side of the equation
• Use inverse operations for simplification

In this problem, after substituting y into the second equation, we obtained:
x - x = 8, which led to:
0 = 8.
This result told us there was no solution.
Understanding the mechanics of these steps is crucial for solving more complex systems.

System of Linear Equations

A system of linear equations is a set of equations with multiple variables. The solutions are the points where the equations intersect.

We commonly use:

• Substitution Method
• Elimination Method
• Graphical Method

to solve these systems.
Depending on the system:

• It can have a unique solution (intersect at one point).
• Infinitely many solutions (coincident lines).
• No solution (parallel lines).

In the given exercise, the system was found to have no solution, indicating parallel lines.
It's essential to identify the type of system and choose the appropriate method for solving it.