Problem 13 - Algebra A Combined Function

Question

Solve each system. $$ \left\\{\begin{array}{l} 6 x-5 z=17 \\ 5 x-y+3 z=-1 \\ 2 x+y \quad=-41 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = -3\), \(y = -35\), \(z = -7\).

Step 1: Rearrange Equations

Start by taking the given system of equations and rearranging them for easier substitution or elimination later.1. \(6x - 5z = 17\)2. \(5x - y + 3z = -1\)3. \(2x + y = -41\)

Step 2: Solve for y in Equation 3

Isolate \(y\) in the third equation:\[2x + y = -41\]Subtract 2x from both sides:\[y = -41 - 2x\]

Step 3: Substitute y in Equation 2

Substitute the expression for \(y\) from Step 2 into the second equation:\[5x - (-41 - 2x) + 3z = -1\]This simplifies to:\[5x + 41 + 2x + 3z = -1\]Combine like terms:\[7x + 3z = -42\]

Step 4: Express x in Terms of z from Equation 1

Rearrange the first equation to solve for \(x\):\[6x - 5z = 17\]Add 5z to both sides:\[6x = 5z + 17\]Then divide by 6:\[x = \frac{5z + 17}{6}\]

Step 5: Substitute x in the Simplified Equation from Step 3

Substitute the expression for \(x\) from Step 4 into the equation obtained in Step 3:\[7\left(\frac{5z + 17}{6}\right) + 3z = -42\]Multiply through by 6 to clear the fraction:\[7(5z + 17) + 18z = -252\]Expand and simplify:\[35z + 119 + 18z = -252\]Combine like terms:\[53z + 119 = -252\]

Step 6: Solve for z

Solve the equation obtained in the previous step for \(z\):\[53z = -252 - 119\]\[53z = -371\]Divide both sides by 53:\[z = -7\]

Step 7: Solve for x using z

Substitute \(z = -7\) back into the expression for \(x\) found in Step 4:\[x = \frac{5(-7) + 17}{6}\]\[x = \frac{-35 + 17}{6}\]\[x = \frac{-18}{6}\]\[x = -3\]

Step 8: Solve for y using x

Use the value of \(x = -3\) in the expression for \(y\) from Step 2:\[y = -41 - 2(-3)\]\[y = -41 + 6\]\[y = -35\]

Step 9: Verify the Solution

Substitute \(x = -3\), \(y = -35\), and \(z = -7\) into the original equations to verify the solution is correct:1. \(6(-3) - 5(-7) = -18 + 35 = 17\)2. \(5(-3) + 35 + 3(-7) = -15 - 21 = -1\)3. \(2(-3) - 35 = -6 - 35 = -41\)All equations are satisfied, confirming the solution is correct.

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving systems of equations, where you solve one of the equations for one variable and then substitute this expression into the other equations. This process helps to reduce the number of variables in the remaining equations, facilitating easier solutions.
Let's explore the steps involved in using the substitution method:

• Identify which equation in the system can be easily rearranged to express one variable in terms of another.
• Solve this equation for the chosen variable.
• Substitute this expression into the other equations in the system.
• Simplify the resulting equations and solve for the remaining variables.

In our example, this method was highlighted particularly in Step 2 where the expression for \(y\) was found from the third equation. We then used this expression in Step 3, substituting it into the second equation. This reduced the complexity of the system, making it straightforward to resolve the remaining variables.

Elimination Method

The elimination method, also known as the addition or elimination process, is another popular approach to solve systems of equations. It involves adding or subtracting equations to eliminate one of the variables. This method simplifies the system and can effectively handle equations that don't lend themselves easily to substitution.
Here’s how you can apply the elimination method:

• Align all equations vertically by variables and constants.
• Multiply one or more equations, if necessary, so that the coefficients of one of the variables are opposites.
• Add or subtract the equations to eliminate one variable.
• Solve for the remaining variable(s).
• Use back substitution to find the eliminated variable(s).

In the solved system, although the main technique used was substitution, eliminating variables was a part of the procedural design seen in Steps 5 and 6. These steps manipulated the equations such that one of the variables, \(z\), could be solved, simplifying the rest of the process.

Linear Algebra

Linear algebra serves as the overarching mathematical structure that simplifies systems like these. It provides a theoretical framework through which systems of linear equations can be approached and solved using various methods like matrices or vector spaces.
Core elements in linear algebra that are related to solving systems of equations include:

• Understanding linear combinations and dependencies, where equations are considered as linear combinations of variables.
• Matrix representations which transform systems of equations into a more manipulable algebraic form.
• Row reduction techniques which relate directly to elimination methods, allowing simplification through operations similar to Gaussian elimination.

Linear algebra makes these systems not only solvable but also understandable in terms of vector spaces and matrix transformations, providing deeper insights into the nature of solutions, such as the existence and uniqueness of solutions. In practice, many times, computer algorithms implement advanced linear algebra techniques to solve very similar problems faster.