Problem 108 - Algebra and Trigonometry

Question

Consider the following equations: \(\left\\{\begin{array}{l}{5 x-2 y-4 z=3} \\ {3 x+3 y+2 z=-3}\end{array}\right.\) Eliminate \(z\) by copying Equation \(1,\) multiplying Equation 2 by \(2,\) and then adding the equations.

Step-by-Step Solution

Answer

The equation after eliminating z is \(11x + 4y = -3\).

Step 1: Setup the Equations

Start by writing out the two equations as given in the problem: \[\begin{cases}5x-2y-4z=3 \3x+3y+2z=-3\end{cases}\]

Step 2: Multiply the Second Equation

Multiply the second equation by 2 so that we can later cancel out \(z\) by adding the two equations together. This results in: \[6x+6y+4z=-6\]

Step 3: Add the Equations Together

Add the first equation (Step 1) and the altered second equation (Step 2) together:\[(5x-2y-4z) + (6x+6y+4z) = 3 + (-6)\]When you add these equations together, \(z\) will cancel out because \(-4z + 4z = 0\). This leaves you with:\[11x + 4y = -3\]

Key Concepts

elimination method

The elimination method is a fantastic tool when working with systems of linear equations. The goal of this technique is to eliminate one of the variables by combining the equations. This makes it possible to solve for the remaining variables one at a time. In the given exercise, we're dealing with a system of two equations:

• Equation 1: \(5x - 2y - 4z = 3\)
• Equation 2: \(3x + 3y + 2z = -3\)

The steps begin by manipulating these equations to eliminate one variable, here it's the \(z\). We do this by making the coefficients of \(z\) opposites, which means when we add the equations, \(z\) cancels out.
If executed correctly like in the steps provided, this method is efficient and direct, making it a favorite among students solving linear systems.

linear equations

A linear equation is a mathematical statement that creates a straight line when graphed. It is also key in forming systems of equations like the ones in this exercise. Linear equations typically have constants and coefficients applied to variables which are to the first degree.
Each equation of the form \(Ax + By + Cz = D\) represents a plane in three-dimensional space. By finding solutions to the system of linear equations, you're essentially finding a point where the lines (or planes, in three dimensions) intersect.
In this particular case, by eliminating the \(z\), we reduce two linear equations in three variables to an equation in two variables, simplifying the problem significantly. Understanding linear equations formulates the backbone for tackling more complex algebraic tasks.

algebraic manipulation

Algebraic manipulation involves the use of various algebraic techniques to simplify, rearrange, or rewrite equations. It's like a mathematical toolkit that you reach for when you need to clean up or clarify an equation or system.
In the context of the exercise, the main algebraic manipulations include:

• Multiplying Equation 2 by 2, allowing the coefficients of \(z\) in both equations to cancel each other out upon addition.
• Adding equations in a way that eliminates one of the variables entirely. This simplifies the system into something more easily solvable.

These manipulations are vital because they transform complex systems into more manageable forms, paving the way towards the solution. Perfecting these skills is crucial for advancing in algebra and other areas of mathematics.