Problem 16 - Algebra and Trigonometry

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{aligned}-4 x+12 y &=0 \\ 12 x+4 y &=160 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has a unique solution: \((12, 4)\).

Step 1: Write the System of Equations

The provided system of equations is: \[-4x + 12y = 0\]\[12x + 4y = 160\]These are two linear equations with two unknowns, \(x\) and \(y\).

Step 2: Simplify the First Equation

Divide every term in the first equation by \(-4\) to simplify it.\[-4x + 12y = 0\] becomes \[x - 3y = 0\] after dividing by \(-4\).

Step 3: Solve the First Equation for x

Express \(x\) in terms of \(y\) using the equation \[x - 3y = 0\].Rearrange to find: \[x = 3y\].

Step 4: Substitute into the Second Equation

Replace \(x\) with \(3y\) in the second equation, \(12x + 4y = 160\).This gives: \[12(3y) + 4y = 160\] which simplifies to \[36y + 4y = 160\].

Step 5: Simplify and Solve for y

Simplify \[36y + 4y = 160\] to \[40y = 160\].Divide both sides by 40 to find \(y\): \[y = 4\].

Step 6: Solve for x using y = 4

Use the result \(y = 4\) in the equation \(x = 3y\).Substitute \(y = 4\) to get \(x = 3(4) = 12\).

Step 7: Write the Solution as an Ordered Pair

The values \(x = 12\) and \(y = 4\) can be written as an ordered pair: \((12, 4)\). This ordered pair is the unique solution to the system of equations.

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. For example, we often work with two linear equations like in our original exercise:

• \(-4x + 12y = 0\)
• \(12x + 4y = 160\)

This system presents two equations with two variables, \(x\) and \(y\). Solving such a system means finding a set of values for these variables that satisfy both equations simultaneously.
There are different methods to solve a system of equations, such as graphing, substitution, or elimination. In this case, we use the substitution method, which involves substituting one variable in terms of another to simplify the problem.

Solution of a System

When solving a system of equations, our goal is to determine the values of the variables that satisfy all equations in the system.
These solutions can fall into three categories:

• Unique Solution: A specific value for each variable. Our exercise concludes with a unique solution: \(x = 12\) and \(y = 4\), which can be expressed as the ordered pair \((12, 4)\).
• Infinitely Many Solutions: When the equations are dependent, meaning they represent the same line, every point on the line represents a solution.
• No Solution: When the equations are inconsistent, like parallel lines that never intersect, meaning there are no common solutions.

The method of substitution often leads directly to a unique or clearly identifiable solution.

Ordered Pairs

Ordered pairs are a fundamental concept in solving systems of equations. An ordered pair is a set of two numbers, typically written as \((x, y)\), which represents a specific point in a coordinate plane.
Once we've found a solution to a system of equations, we express the values of \(x\) and \(y\) as an ordered pair. This pair shows the precise intersection point of the two lines represented by the equations when graphed.
In the original exercise, the ordered pair \((12, 4)\) was the result upon solving the system, meaning that this specific combination of \(x\) and \(y\) satisfies both equations.

Substitution Method

The substitution method is a technique to solve a system of equations by substituting one equation into another. This process involves several clear steps:

• Solve one equation for one variable: In our case, we started by simplifying the first equation \(-4x + 12y = 0\) and expressed \(x\) in terms of \(y\): \(x = 3y\).
• Substitute this expression into the other equation: We replaced \(x\) in the second equation \(12x + 4y = 160\), simplifying to a single-variable equation \(40y = 160\).
• Solve for the remaining variable: We solved for \(y\) to find \(y = 4\).
• Back-substitute to find the other variable: We used \(y = 4\) to find \(x = 12\).

The substitution method is highly effective, especially when one of the equations is easy to manipulate or simplify.