Question

Solve the system of equations. $$\left\\{\begin{aligned} 4 x+3 y &=7 \\ x-2 y+z &=0 \\\\-2 x+4 y-2 z &=13 \end{aligned}\right.$$

Short answer

The given system of equations has no solutions. The inconsistency becomes apparent when eliminating variables through combining equations.

Step-by-step solution

Preparation

Rewrite the system of equations as follows: \[ \begin{align*} (1) & : 4x + 3y = 7 \ (2) & : x - 2y + z = 0 \ (3) & : -2x + 4y - 2z = 13 \end{align*} \]

Elimination

Multiply equation (2) by 2 and add it to equation (3) to eliminate \(x\). This gives: \[ (2) * 2 : 2x - 4y + 2z = 0 \] \[ (3) + (2) * 2 : 0x + 0y = 13 \] This is not possible, so the system of equations has no solution.

Conclusion

As it turns out, these three equations are inconsistent; there's no set of values for \(x\), \(y\), and \(z\) that will satisfy all three at the same time. This conclusion is reached at the moment when all variables are eliminated and only a false statement remains. Hence the system has no solutions.

Key concepts

Elimination Method

The elimination method is a technique used in solving systems of linear equations, where the goal is to remove one or more variables to find a solution. To understand this strategy, imagine you have several balance scales, each with weights representing the variables and constants in an equation. By carefully adding or subtracting entire scales, you aim to balance them out in such a way that one type of weight (or 'variable') disappears. It’s a little like solving a puzzle: you must determine the right combination of moves that simplifies the problem.

Applied to the given exercise, we attempted to eliminate the variable by multiplying the second equation by 2 and adding it to the third equation. While ideally, this would remove the variable and produce a simpler equation that can be solved, in this case, we reached an equation with all variables eliminated, leading to an impossible statement (equation (3) after addition: 0x + 0y = 13). This is a strong indicator that the system may be inconsistent, which ties into the concept of an 'inconsistent system'.

Inconsistent System

An inconsistent system of equations is like trying to find a unicorn—it’s a mythical scenario that can't exist in reality. More technically, it’s a set of equations that has no solution because there are no possible values of the variables that would satisfy all the equations simultaneously. When you line up all the equations, you'll find they're pulling in different directions and can never meet at a single point (or line, or plane).

In the case of our exercise, multiplying the second equation by 2 and adding it to the third equation stripped all variables, leaving an absurdity: '0 equals 13'. This is a hallmark of an inconsistent system—akin to saying you've caught a shadow—a clear indication you’re on a chase that will yield no solution.

Algebraic Reasoning

Algebraic reasoning is the Sherlock Holmes of math, involving logical deduction to investigate properties, relationships, and the solvability of algebraic expressions and equations. You look for clues in the form of coefficients and constants, use 'operations' as investigative tools, and apply laws—like distributive or associative—to unravel the mysteries of the algebraic puzzle. It's the process of thinking through the steps required to simplify or solve algebraic problems.

During our exercise, we used algebraic reasoning to decide on multiplying equation (2) by 2—doubling our 'clues' to make a more potent deductive move—and finally recognized that the resulting impossible statement (0x + 0y = 13) could be no other than a dead end. This reasoning leads us to the only possible conclusion of 'no solution'.

No Solution

When you are told there is 'no solution', it's not a concession of defeat; it's an important mathematical conclusion. In the context of solving systems of equations, 'no solution' means that there is no set of numbers that you can plug into the variables to make all the equations true at once. It is as if you're being told that there is no treasure at the end of the map—because X does not mark any spot.

Through our algebraic detective work in the exercise, we discovered that after manipulation there is an equation where 0 equals 13, which is impossible: it's a signpost saying there's no treasure to be found. This taught us that sometimes, not finding a solution is as significant as finding one, because it reveals more about the problem itself.