Problem 42 - Algebra and Trigonometry

Question

Solve the system of linear equations. $$ \left\\{\begin{aligned}-4 x-y+36 z &=24 \\ x-2 y+9 z &=3 \\\\-2 x+y+6 z &=6 \end{aligned}\right. $$

Step-by-Step Solution

Answer

This system of equations has no solution.

Step 1: Select and Simplify Equations

Choose two equations to work with initially. Let's work with the first two equations: \(-4x - y + 36z = 24\) and \(x - 2y + 9z = 3\). Multiply the second equation by 4 in order to make elimination of \(x\) easier: \(4x - 8y + 36z = 12\).

Step 2: Eliminate x Variable

Add the two equations from Step 1 together to eliminate \(x\):\[(-4x - y + 36z) + (4x - 8y + 36z) = 24 + 12\]This simplifies to: \(-9y + 72z = 36\). Simplify further to: \(y = 8z - 4\).

Step 3: Substitute y in Third Equation

Use the expression for \(y\) from Step 2 in the third equation: \(-2x + (8z - 4) + 6z = 6\). Simplify to get: \(-2x + 14z - 4 = 6\).

Step 4: Solve for x in Terms of z

Rearrange the equation from Step 3: \(-2x + 14z = 10\). This gives: \(x = 7z - 5\).

Step 5: Solve for z

Substitute the expression \(y = 8z - 4\) and \(x = 7z - 5\) back into the second equation \(x - 2y + 9z = 3\): \[(7z - 5) - 2(8z - 4) + 9z = 3\]Simplify to find \(z\): \( 7z - 5 - 16z + 8 + 9z = 3\) which simplifies to \(-4 = 0\), indicating a contradiction. This means there is no solution.

Key Concepts

Elimination Method

The elimination method is a technique used to solve systems of linear equations, where we aim to eliminate one variable by manipulating the equations. This is because working with fewer variables can often make the problem easier to solve. Imagine trying to untangle a couple of tightly woven threads: getting rid of one simplifies the mess.

To start, you select two equations from the system. The goal is to add or subtract these equations so that one of the variables is eliminated. You might need to multiply one or both equations by certain numbers to align terms correctly. In our example, we chose to eliminate the variable \(x\). Here's a quick guide:

• Select pairs of equations.
• Multiply if necessary to align coefficients of the chosen variable.
• Add or subtract the equations to eliminate that variable.

After eliminating one variable, you solve the resulting simpler system for the remaining ones. This reduces complexity and helps us find solutions faster.

No Solution

When solving a system of equations, the outcome might be that there is no solution. This indicates that the equations in the system cannot all be true simultaneously. Imagine trying to place three points in space such that all three are aligned perfectly on different, non-overlapping lines—it's impossible.

In the elimination method, reaching a logical inconsistency, like ending up with a false statement (as in \(-4 = 0\)), signals that no solution exists. It's a red flag for parallel lines that never intersect, or in three-dimensional terms, planes that don't share a common point. Here’s how to identify systems with no solutions:

• Look for contradictions, like impossible equations (\(-4 = 0\)).
• Notice if the system reduces to parallel lines.
• Listen for signs of inconsistency during substitution or elimination.

This outcome suggests that the system of equations represents geometries that do not meet or intersect at any single point.

Substitution Method

The substitution method offers another approach to solving systems of linear equations. It involves solving one of the equations for one variable and substituting that expression into another equation. This technique is like gradually finding your way out of a maze by building on each previous step.

In the original example, after finding an expression for \(y\) \(y = 8z - 4\), it was substituted back into another equation. This method is ideal when equations are already nearly solved for one variable. Here's the process:

• Pick an equation and solve it for one variable.
• Substitute that expression into other equations.
• Solve the resulting simpler system.

The substitution method can sometimes reveal insights into relationships between variables, nudging us closer to solutions step by step. It's all about consistent progress and building upon previous calculations.