Problem 17 - College Algebra

Question

Solve each system. $$\begin{aligned}3(2 x+y)+5 z &=-1 \\\2(x-3 y+4 z) &=-9 \\\4(1+x) &=-3(z-3 y)\end{aligned}$$

Step-by-Step Solution

Answer

The solutions are \(x = -1\), \(y = 2\), and \(z = -1\). It's important to verify these back into the original equations to ensure they satisfy all of them.

Step 1: Simplify the given equations

Simplify the given equations as follows: \n First equation simplifies to \(6x + 3y + 5z = -1\) \n Second equation simplifies to \(2x - 6y + 8z = -9\) \n The third equation simplifies to \(4 + 4x = -3z +12y\)

Step 2: Rearrange the equations in terms of x, y, and z

Rearrange the simplified equations as follows: \n The first equation becomes \(6x = -3y - 5z - 1\) \n Divide the whole equation by 6 to get \(x = -\frac{1}{2}y - \frac{5}{6}z - \frac{1}{6}\) \n The second equation becomes \(2x = 6y - 8z + 9\) \n Divide the whole equation by 2 to get \(x = 3y - 4z + \frac{9}{2}\) \n For the third equation, first bring the terms involving z to the LHS so it becomes \(4 + 4x + 3z = 12y\), then divide the whole equation by 12 to get \(y = \frac{1}{3}x + \frac{1}{4}z + \frac{1}{3}\)

Step 3: Equate the equations for x and y

Equate the two equations for x and solve for y: \n-\(\frac{1}{2}y - \frac{5}{6}z - \frac{1}{6} = 3y - 4z + \frac{9}{2}\) \n Express equation for y in terms of other variables: \n\(\frac{1}{3}x + \frac{1}{4}z + \frac{1}{3}\)

Step 4: Solve for the variables

Solve for the variables by substitution or elimination method: \n Substitute the expressions for x and y in terms of z from Step 3 into one of the original equations. This creates an equation only in terms of z. Solve this to get the value of z. \n Then substitute the value of z back into the equations for x and y obtained in step 3 to solve for the other two variables.

Key Concepts

Simplifying Equations

Simplifying equations is like untangling a knot, making everything easier to handle. In our given exercise, we start with equations packed with parentheses and factors. Our task is to

• eliminate parentheses using distributive laws
• combine like terms if possible

This gives us cleaner and more manageable equations. For instance, the first equation after simplification became \(6x + 3y + 5z = -1\). By removing the parentheses and factoring elements meticulously, we startingly reveal the core structure of each equation. This step sets the stage for using more advanced techniques to find the solution.

Substitution Method

The substitution method is an elegant way to solve a system of equations by expressing one variable in terms of another. Like swapping one piece for another in a puzzle, it makes solving easier. Once the equations are simplified, you can

• solve one equation for one variable
• use this expression to replace the variable in the other equation

In this exercise, into one of our simpler equations, we substituted expressions for x and y derived from other arrangements. The substitution enabled us to achieve a single equation with just one unknown variable. This method simplifies the solving chain greatly.

Elimination Method

The elimination method aims to "cancel out" one variable at a time, making the path to a solution clear. It's like methodically dismantling a tower, piece by piece, until you uncover the core foundation. You align the simplified equations and then

• add or subtract equations to remove one variable completely
• repeat this process if necessary until only one variable remains

While we focused more on substitution in this problem, elimination could have been another path forward. By eliminating one variable, you gradually expose the values of the others, leading you methodically to the solution.

Algebraic Manipulation

Algebraic manipulation transforms equations into their most workable forms. By rearranging and altering the equations, we make them into a user-friendly format. Here's how it helps:

• By rearranging terms to isolate variables on one side
• Scaling whole equations to simplify variable comparisons

The algebraic maneuvering in our exercise involved rearranging terms such as \(4 + 4x = -3z + 12y\) to solve efficiently. Clear manipulation paved the way for direct substitutions and eventual solutions. With each step, the complex gets transformed into simplicity, leading us surely toward the answer.