Problem 50 - Precalculus

Question

Explain the mistake that is made. Solve the system of equations. Equation (1) \(x+3 y+2 z=4\) Equation (2): \(3 x+10 y+9 z=17\) Equation (3): \(7 y+7 z=17\) Solution: Multiply Equation (1) by \(-3, \quad-3 x-9 y-6 z=-12\) \(\begin{array}{lr}\text { Equation }(2): & \frac{3 x+10 y+9 z=}{y+3 z=5} \\\ \text { Add. } & y+3 z=5\end{array}\) Multiply Equation (1) by \(-2 . \quad-2 x-6 y-4 z=-8\) \(\begin{array}{lr}\text { Equation }(3): & \frac{2 x+7 y+7 z=17}{y+3 z=9} \\\ \text { Add. } & y+3 z=9\end{array}\) Solve the system of two equations. \(y+3 z=5\) \(y+3 z=9\) Infinitely many solutions. Let \(z=a,\) then \(y=5-3 a\) Substitute \(z=a\) and \(y=5-3 a\) into \(x+3 y+2 z=4\) Equation (1). \(x+3(5-3 a)+2 a=4\) Eliminate parentheses. \(x+15-9 a+2 a=4\) Solve for \(x\) The answer is \(x=7 a-11, y=5-3 a,\) and \(z=a\) This is incorrect. There is no solution. What mistake was made?

Step-by-Step Solution

Answer

The main error is incorrect math manipulations and equation transformations leading to flawed terms.

Step 1: Express Equation (1) for Elimination

Multiply Equation (1), \(x + 3y + 2z = 4\), by \(-3\) to create an expression for elimination: \[-3x - 9y - 6z = -12\].

Step 2: Incorrect Substitution in Equation (2)

The mistake is here: the solution incorrectly shows Equation (2) after multiplying Equation (1). It shows:\[\text{Equation (2): } \frac{(3x + 10y + 9z) = (y + 3z = 5)}\]This is incorrect as it mislabels terms and equations, leading to an incorrect system.

Step 3: Recognize Incorrect Substitution for Equation (3)

The same mistake occurs when substituting incorrectly into Equation (3). It should have been handled correctly without transforming it via incorrect math manipulation. The problem derived:\[\text{Equation (3): } \frac{(2x + 7y + 7z = 17)}{(y + 3z = 9)}\] Instead, the term manipulations are incorrect.

Step 4: Solve Correctly

Equation (3) is correctly expressed as \(7y + 7z = 17\), which means: \[y + z = \frac{17}{7}\].Combining this correctly with other equations using the substitution method would have shown that the previous elimination attempt was flawed.

Step 5: Conclusion on the Equations

Identify the mistake: Incorrect manipulation and equation reduction led to incorrect solutions \(y + 3z = 5\) and \(y + 3z = 9\) suggesting a wrong infinite solutions scenario. Revisiting valid substitutions and alignment would have shown no real solution due to inconsistencies.

Key Concepts

Elimination Method

The elimination method is a technique used to solve systems of linear equations. The main idea is to eliminate one of the variables by adding or subtracting equations, ultimately simplifying the system so that one can easily find the values of the remaining variables. Consider a system of equations where each equation can be manipulated to align terms for the effective cancelation of variables.
Here's how it works in detail:

• Multiply one or both of the equations so that the coefficients of one of the variables are opposites.
• Add or subtract these new equations to eliminate one variable.
• With one variable removed, solve for the remaining variable(s), and gradually find the value of all unknowns.

This method requires careful handling to ensure each step is executed precisely, aligning coefficients correctly to avoid errors in deduction, which can lead to incorrect solutions.

Incorrect Substitution

Substitution involves solving one equation for one variable and then using this to substitute into another equation. This is typically done after simplifying the system through elimination. However, a critical mistake occurs when incorrect substitution is applied, leading to false conclusions.
Common errors during substitution include:

• Mislabeling or transforming terms incorrectly when rearranging equations.
• Confusion in managing signs, which can drastically alter the results.
• Failing to check if the manipulation actually simplifies the equation towards a valid solution.

In the provided exercise, substitution was applied incorrectly, leading to mismatched terms like incorrect coefficients, resulting in an inconsistent system and falsely implying possible infinite solutions.

No Solution

When solving a system of equations, sometimes you can encounter a situation where no solution exists. This happens when the equations describe parallel lines, meaning they never intersect. In mathematical terms, this is expressed through inconsistencies that make it impossible to satisfy all equations simultaneously.
Indicators of no solution include:

• Equations reducing to a contradiction, e.g., suggesting something like "0 = 5".
• Variables canceling out, leaving behind incompatible constants.

In the given exercise, after observing incorrect manipulations, the conclusions drawn falsely suggested solutions. However, correct analysis confirms there is no point satisfying all equations due to inherent discrepancies stemming from flawed substitution and elimination steps.