Problem 69 - Precalculus

Question

Employ a graphing calculator to solve the system of equations. $$\begin{array}{rr} 0.2 x-0.7 y+0.8 z= & 11.2 \\ -1.2 x+0.3 y-1.5 z= & 0 \\\ 0.8 x-0.1 y+2.1 z= & 6.4 \end{array}$$

Step-by-Step Solution

Answer

Use a graphing calculator to input and solve the equations for \(x, y, z\), then verify by substitution.

Step 1: Enter the Equations

Power on your graphing calculator and enter the equations into the simultaneous equation solver. Input each equation carefully as follows: \(0.2x - 0.7y + 0.8z = 11.2\), \(-1.2x + 0.3y - 1.5z = 0\), and \(0.8x - 0.1y + 2.1z = 6.4\). Ensure you correctly input the coefficients of \(x\), \(y\), and \(z\) and the constant term on the right.

Step 2: Solve the System

Use the calculator function to solve the system of equations. After entering the equations, follow the calculator prompts to compute the solution. The solution will appear as the values for \(x\), \(y\), and \(z\) that satisfy all three equations.

Step 3: Verify the Solution

The graphing calculator will provide a solution, typically in the form \((x, y, z)\). To verify, substitute these values back into the original equations to check if they satisfy each equation, ensuring that any rounding does not significantly alter the results.

Key Concepts

graphing calculator

A graphing calculator is a versatile tool that can do more than just plot graphs. It is designed to handle both arithmetic calculations and sophisticated tasks like solving systems of equations.

When dealing with a system of equations like:

• \(0.2x - 0.7y + 0.8z = 11.2\)
• \(-1.2x + 0.3y - 1.5z = 0\)
• \(0.8x - 0.1y + 2.1z = 6.4\)

using a graphing calculator simplifies the complex arithmetic involved. You can input the equations into the calculator's simultaneous equation solver function.

This feature enables the entry of multiple equations and calculates their intersection points—which, for equations, are the solutions. Modern graphing calculators offer a prompt-driven input system, making it simple to enter coefficients and constants from the equations accurately.

They are built to minimize human error while dealing with multi-variable problems, ensuring you find precise solutions.

simultaneous equations

Simultaneous equations are sets of equations with multiple variables. The objective is to find common solutions for these variables that satisfy each equation simultaneously. In the given problem, we have three variables, namely, \(x\), \(y\), and \(z\).

• Each equation represents a plane in three-dimensional space.
• The solution \((x, y, z)\) is the point where all the planes intersect.

By using algebraic methods, you typically rewrite the equations to manipulate and eliminate variables step-by-step. However, this can be time-consuming and complex when dealing with more than two variables. This is why graphing calculators are vital for efficiently solving them.

These equations can arise in various practical applications, like in engineering for circuit analysis, in economics for market equilibrium, and in physics for force resolution.

solution verification

Solution verification involves checking that the solution values for \(x\), \(y\), and \(z\) obtained from the calculator are correct. This crucial step ensures that our solution is both valid and accurate.

• First, substitute these values back into each of the original three equations.
• Verify that each equation holds true with these values.

For example, if your calculator gives a solution for the system as \((x, y, z)\), place these values into:

• \(0.2x - 0.7y + 0.8z = 11.2\)
• \(-1.2x + 0.3y - 1.5z = 0\)
• \(0.8x - 0.1y + 2.1z = 6.4\)

This means checking if both sides of each equation equal after substitution. Minor discrepancies might occur due to rounding cases in the calculator. If substantial discrepancies arise, you should re-evaluate your solution entry and calculations. Always conclude the verification step to ensure full confidence in the system's solution.