Problem 39 - Algebra and Trigonometry

Question

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. \(\left\\{\begin{array}{l}{\frac{x}{4}-\frac{y}{4}=-1} \\ {x+4 y=-9}\end{array}\right.\)

Step-by-Step Solution

Answer

(-5,-1) is the unique solution to the two equations

Step 1: Express each equation in standard form

First, both equations in the system are rearranged into the form A*X + B*Y = C: \[ x- y =-4 \] and \[ x+4y =-9 \]

Step 2: Solve the System by substitution

Express x from the first equation as \( x = y - 4 \), substitute x in second equation, we get \[ y - 4 + 4y = -9 \] which simplifies to \[ 5y = -5 \], solving for y gives \( y = -1 \)

Step 3: Find the Value of X

Substitute \( y = -1 \) in the first equation to find x, We get \( x = -1 - 4 \) so, \( x= -5 \)

Step 4: Analyze the Result

Since there is a unique solution, it means that the system of equations has a single intersection point (-5,-1)

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It's especially useful when the equations involve linear terms. Here's how it works: you solve one of the equations for one variable and then substitute that expression into the other equation. This transforms the system into a single equation with one unknown, which you can solve normally.
The substitution method involves a few straightforward steps:

• Solve one of the equations for a variable. This can be x or y depending on which makes simplification easier.
• Substitute this expression into the other equation. This substitution reduces the original system into one equation.
• Solve this new equation for its variable. Typically, this step is just basic algebra.

This method is particularly effective for smaller systems or when one equation can be easily rearranged. Understanding the substitution method is crucial for solving linear systems efficiently and accurately.

Standard Form Equations

Standard form equations are mathematical expressions of linear equations typically arranged in the form \( A \cdot x + B \cdot y = C \). The coefficients \( A \), \( B \), and the constant \( C \) are integers. This form helps in visualizing the equation and makes it easier to manipulate and solve.
In practice, converting equations to standard form requires reorganizing terms and often simplifying fractions. Notice in our exercise, the equations are rearranged to:

• \( x - y = -4 \)
• \( x + 4y = -9 \)

The standard form helps identify the coefficients of x and y directly. Transitioning to this form is an important initial step, especially when further methods like substitution or elimination are to be applied. It amplifies clarity and prepares the equations for systematic solving.

Unique Solution

A unique solution in a system of equations means that there is exactly one set of values for the variables that satisfies both equations simultaneously. Geometrically, this represents the point of intersection between two lines.
In our exercise, by solving the equations using substitution, we reached a solution of \( x = -5 \) and \( y = -1 \). This indicates that the lines represented by these equations intersect at the point \((-5, -1)\).
Having a unique solution suggests that the lines cross at a single distinct point, showing that the system is consistent and independent. This outcome is crucial for understanding how two equations interact and ensuring that the solution is both valid and complete.

Consistent System

A consistent system of equations is one in which a solution exists; that means the lines either intersect at one point or are the same line (infinitely many solutions). In this context, our system has a unique intersection point, thus it's a consistent system.
To analyze consistency, observe whether the solution process yields a valid outcome. Here, each step from rearranging to substitution checks out, confirming a real intersection point. This validation shows the system isn’t contradictory.
Consistency is fundamental in linear algebra as it ensures that the equations lead to a logical and solvable outcome. It's what we aim for when resolving systems using algebraic methods.