Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{r}7 x+8 y=16 \\ -14 x-16 y=-4\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is an infinite set of points along the line \(y = 2 - \frac{7x}{8}\). This result is accurately represented by the single equation from step 2, and it offers an infinitely many solutions for the original system of equations.

Step 1: Simplify the System

In this system of equations, notice that the whole second equation is the result of multiplying the first equation by \(-2\). Hence the second equation is dependent on the first. Let's write the simplified system, removing the second equation: \(\left\{\begin{array}{r}7 x+8 y=16 \ 0=0\end{array}\right.\)

Step 2: Solve the Remaining Equation

Now, we only have one equation left. Let's solve the remaining equation for \(y\). \[y = \frac{16 - 7x}{8}\] simplifying we get \(y = 2 - \frac{7x}{8}\). This is a linear equation, with \(x\) as the independent variable and \(y\) as dependent variable

Step 3: Interpret the Results

Since the second equation in the original system was dependent on the first, a solution to the first equation is also a solution to the original system. This tells us that we have an infinite number of solutions to the system presented, and all of these solutions lie on the line given by \(y = 2 - \frac{7x}{8}\). These solutions can be verified and visualized using a graphing utility.

Key Concepts

Elimination Method

The elimination method is a popular technique used for solving systems of linear equations. The main goal of this method is to eliminate one of the variables, allowing you to solve for the other variable with ease.
In the elimination method, you align the equations in parallel order, like this:

• The coefficients of either variable in both equations are altered to allow subtraction or addition, effectively removing that variable from one of the equations.
• This involves adding or subtracting equations from each other after multiplying by necessary constants.
• The resulting equation will have only one variable left, making it easier to solve.

In this exercise, multiplication was used to demonstrate that the two given equations are dependent, thus simplifying the system.

Dependent Equations

Dependent equations occur when one equation in a system is a multiple or a linear combination of another. Essentially, no new information is obtained from the secondary equations. In our exercise:
- After observing that the second equation was merely the first multiplied by a constant, the equation became redundant.
- This redundancy is often not evident without comparison or simplification.
- The presence of dependent equations confirms that the solutions of one equation apply to the others and lie along the same line in the coordinate plane.

Infinite Solutions

Infinite solutions arise when the equations in a system describe the same line or curve; they have all their solutions in common.
Recognizing equations that are dependent hints at the possibility of infinite solutions. Key points include:

• Since the second equation was a version of the first, it didn't offer new intersections or solutions.
• The infinite solutions are represented graphically by the overlap of the same line, confirming the system's consistency.
• In our scenario, every solution on the line given by the simplified equation is also a solution for the entire original system.

This shared line effectively holds an infinite number of points, each one satisfying both original equations.

Linear Equation

A linear equation is an equation that makes a straight line when graphed. It’s expressed in the format of: \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
Linear equations are characterized by:

• Their graph is always a straight line.
• Their solutions can be easily represented in a linear algebraic form.
• Each point on the line represents a pair of \(x\) and \(y\) that holds the equation true.

In the exercise, solving the system resulted in the linear equation: \(y = 2 - \frac{7x}{8}\). It reveals all potential solutions of the system as points along this line in a two-dimensional plane.