Question

Solve the system of equations. $$\left\\{\begin{aligned} 2 x+y-z &=13 \\ x+2 y+z &=2 \\ 8 x-3 y+4 z &=-2 \end{aligned}\right.$$

Short answer

The solution of the system of equations is x = 3, y = 4.5, and z = 2.5.

Step-by-step solution

Multiply equations to align coefficients

Multiply the first equation by 4 and the second equation by 8, with the goal of aligning coefficients with the third equation for easy elimination. This results in: \[ \begin{aligned} 8x+4y-4z &=52, \ 8x+16y+8z &=16, \ 8x-3y+4z &=-2. \end{aligned}\]

Perform the algebraic elimination

Subtract the newly obtained first equation from the third and the second equation from the first. We get: \[ \begin{aligned} -7y+8z &=-54, \ 8y &=36. \end{aligned}\]

Find the variable y

We can solve for y from the second equation which simplifies to: \[ y = \frac{36}{8} = 4.5 \].

Find the rest of the variables

Substitute y = 4.5 into the equations -7y+8z = -54, 2x+y-z =13. After some algebraic simplification, we obtain that z = 2.5, x = 3.

Key concepts

Algebraic Elimination Method

When solving systems of linear equations, the algebraic elimination method is a strategic approach to reduce complexity and isolate variables. This method involves manipulating equations to cancel out one or more variables, making it easier to find solutions for the remaining variable(s). To employ this technique effectively, one can multiply or divide equations to align their coefficients, which sets up the opportunity to add or subtract equations from one another and eliminate variables.

The process often involves a sequence of steps, each simplifying the system further. For instance, in the given exercise, the first step included multiplying the equations to get the same coefficient for the variable x across all three equations. This sets the stage for subtraction, which leads to the elimination of the x variable. Subsequential steps focus on strategically eliminating other variables until only one is left, which can then be easily solved. After solving for one variable, you utilize back substitution to find the values of the remaining variables, as was demonstrated in the steps of the provided textbook solution.

It's important for students to learn this method as it teaches them about critical thinking and systematic problem-solving, which are essential skills not just in algebra, but in various fields of study.

Linear Equation Systems

A system of linear equations is a set of multiple equations each representing a line, where each line has variables that appear to a power of no higher than one. The goal is to find the values of the variables that satisfy all the equations simultaneously. These systems can have one solution (intersecting at a single point), no solution (parallel lines that never intersect), or infinitely many solutions (overlapping lines).

Understanding the fundamentals of linear equation systems is crucial for students, as these systems underpin many areas of mathematics and apply to real world scenarios like predicting expenses, understanding trends, or even programming. In our example, we have a system of three equations with three unknowns, a scenario where the algebraic elimination method is highly suitable for finding the unique solution for such a system.

Linear systems can be represented in several forms such as standard form, slope-intercept form, or matrix form. Standard form, which is shown in the exercise, is when equations are written as Ax + By + Cz = D. This form is especially useful for algebraic elimination as it aligns variables for easy manipulation.

Algebraic Manipulation

The term algebraic manipulation encompasses various techniques used to rearrange and simplify algebraic expressions and equations. This includes operations such as adding, subtracting, multiplying, or dividing both sides of an equation by the same number, factoring, expanding, and rationalizing expressions.

Grasping algebraic manipulation skills is fundamental to solve not just linear systems but also more complex algebraic problems. It aids students in understanding the relationships between variables and how to manipulate those relationships to find solutions. In the context of the exercise, algebraic manipulation was used in multiple steps, including the initial step where equations were multiplied to line up coefficients, and in subsequent steps such as simplifying -7y + 8z = -54 and 8y = 36 to isolate and solve for the variables y and z.

Developing a strong ability in algebraic manipulation also involves recognizing patterns and employing strategies like combining like terms, which can simplify the problem-solving process significantly. Such proficiency not just solves mathematical problems but also builds a foundation for logical reasoning.