Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the system. $$ \left\\{\begin{array}{r} 3 p-q=7 \\ -12 p+4 q=3 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has no solution; it is inconsistent.

Step 1: Analyze the System of Equations

We have two linear equations: 1. \( 3p - q = 7 \) 2. \( -12p + 4q = 3 \). Examine if these two equations can be simplified. Notice that the second equation can be simplified by dividing each term by 4.

Step 2: Simplify the Second Equation

Divide every term of the second equation \( -12p + 4q = 3 \) by 4 to simplify: \( -3p + q = \frac{3}{4} \). The new system of equations is:1. \( 3p - q = 7 \)2. \( -3p + q = \frac{3}{4} \).

Step 3: Add the Equations

Add the simplified equations to eliminate one variable:\[(3p - q) + (-3p + q) = 7 + \frac{3}{4}\]The \( 3p \) and \( -3p \), as well as \( -q \) and \( q \), cancel each other out, leaving:\[0 = 7 + \frac{3}{4}\]This simplifies to say \(0 = \frac{31}{4}\), which is a contradiction.

Step 4: Analyze the Result

The statement \(0 = \frac{31}{4}\) shows a contradiction, meaning that there is no solution to the original system of equations.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra that express relationships between variables using linear expressions. A linear equation is any equation that can be written in the standard form:

• \( ax + by = c \)
• Where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.

In the exercise, we are given two linear equations involving the variables \(p\) and \(q\). Each equation provides a straight line when plotted on a graph. The solution to a system of linear equations is the point where these lines intersect. However, in some cases, the lines may be parallel or coincide, leading to different types of solutions. Understanding the structure of linear equations helps in identifying their solutions and any special characteristics they might exhibit.

Algebraic Simplification

Algebraic simplification is a process used to make equations easier to work with. It involves reducing complexity by combining like terms and performing arithmetic operations to produce a simpler, equivalent equation.
In the example provided, simplification is applied by dividing every term of the equation \(-12p + 4q = 3\) by 4.
This results in a more straightforward equation: \(-3p + q = \frac{3}{4}\). Such simplification helps to align equations to a form that is easier to manipulate, thereby facilitating methods such as substitution or elimination. Simplified equations are easier to compare and solve, especially when dealing with systems of equations.

Contradiction in Systems

A contradiction in systems occurs when two expressions assert opposing conclusions. This is a critical result that indicates there is no possible solution that satisfies both equations simultaneously.
In the process of solving the given system, the equations were added to eliminate variables, leading to a derived equation \(0 = \frac{31}{4}\). This equation is always false, thereby indicating a contradiction.

• If two lines represented by linear equations never meet, it means they are parallel, which also reflects in their algebraic expressions yielding a contradiction when combined.

Identifying contradictions is crucial as it directs us to the conclusion that the system has no solution.

No Solution Systems

No solution systems are characterized by the absence of any intersection between the lines represented by the equations. This occurs when the lines are parallel, which implies they have the same slope but different intercepts.
In our completed solution, the two simplified equations led to a contradiction, proving that the system is a no solution system. When solving linear equations:

• If you encounter a statement like \(0 eq c\) (where \(c\) is a non-zero constant) after simplifying, it indicates parallel lines, signifying no intersection or solution.
• Recognizing no solution scenarios allows learners to understand the graphical interpretation of solutions in linear algebra.

Awareness of no solution systems helps in correctly interpreting and predicting the outcomes of solving real-world problems modeled by linear equations.