Problem 24 - Precalculus with Limits

Question

Solving a System by Elimination In Exercises \(13-30\) , solve the system by the method of elimination and check any solutions algebraically. $$ \left\\{\begin{aligned} 7 x+8 y &=6 \\\\-14 x-16 y &=-12 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system of equations has infinite solutions because adding or subtracting the two equations eliminates both variables, and the resulting equation is always true.

Step 1: Set up the system of equations

The given system of equations is \[ \begin{aligned} 7 x+8 y =6 \\ -14 x-16 y =-12 \end{aligned}\]

Step 2: Elimination

The elimination method requires adding or subtracting the equations in order to cancel one variable. In this case, the second equation could be added to the first one to eliminate \(x\). This results in \(0x = 0\) which simplifies to \(0 = 0\). This is a true statement, which indicates that the original system was dependent and had infinite solutions.

Step 3: Checking the solution algebraically

To confirm that there are indeed an infinite number of solutions, it's required to substitute arbitrary values of \(x\) and \(y\) in both equations. If both equations hold true then the solution is correct.

Key Concepts

Elimination Method

To solve a system of equations like the one given, the elimination method is an effective tool. It involves manipulating the equations to "eliminate" one of the variables, making it easier to solve for the remaining variable. Let's break it down step by step.

• First, align the equations in a format where variables are on one side and constants on the other. For example, both equations should look something like this:\[ \begin{aligned} ax + by = c \ dx + ey = f \end{aligned} \]


• Choose one variable to eliminate. This is usually the variable that allows for easier calculation, or where the coefficients align well for elimination.


• Multiply, if necessary, one or both equations by a number that will make the coefficients of one variable equal in both equations.


• Add or subtract the equations to eliminate one variable. In the given problem, adding the equations cancels out the variable \(x\).

Once a variable is eliminated, you will get an equation like \(0 = 0\), revealing insights about the system's nature, such as whether it has infinite solutions or none.

Dependent System

A dependent system of equations is quite special. It represents multiple equations that describe the same line mathematically.

What does that mean? If plotted on a graph, each equation in a dependent system will overlap exactly, each representing the same line.

• Dependent systems have an infinite number of solutions because any point on the line satisfies all equations in the system.


• When using the elimination method, a dependent system will reveal itself when the resulting equation is a true statement, such as \(0 = 0\).


• Such systems often arise when equations are simple multiples of each other, just like in the given exercise.

Recognizing a dependent system is key when solving systems of equations, as it determines that you do not need a specific solution, but rather understanding that there are countless solutions that satisfy it.

Infinite Solutions

Infinite solutions in a system of equations mean that there is not one unique solution but rather an entire line of possible solutions.

• This occurs in a dependent system where the equations represent the same line.


• If by elimination or substitution the final result is a true statement such as \(0 = 0\), it indicates infinite solutions. This tells us that every combination of \(x\) and \(y\) that satisfies one equation will satisfy the other as well.


• To further confirm infinite solutions, you can substitute several pairs of \(x\) and \(y\) values into the original equations to verify that they hold true.

It's essential when solving equations to recognize when a system has infinite solutions. This understanding can help avoid unnecessary calculations and directly leads to the conclusion that any point on the line shared by the equations is a solution.