Question

Solve the system using the elimination method. \(-2 x-3 y+z=-6\) \(x+y-z=5\) \(7 x+8 y-6 z=31\)

Short answer

The system of equations has infinitely many solutions, as all three planes depicted by the equations intersect along a line.

Step-by-step solution

Choose two equations to eliminate one variable

Let's choose the first and second equations to eliminate 'x'. Because the coefficients of 'x' in both equations are -2 and 1, we can multiply the second equation by 2 to facilitate the elimination. The equations become: \(-2x - 3y + z = -6\)\(2x + 2y - 2z = 10\)

Add the adjusted equations

By adding both equations together, we get \[-2x + 2x - 3y + 2y + z - 2z = -6 + 10\]which simplifies to \[-y - z = 4\]

Choose another pair of equations to eliminate the same variable

Now let's use the second and the third equations to eliminate 'x' again. Here, we can multiply the second equation by 7 and the third by -1 to get\[7x + 7y - 7z = 35\]\[-7x - 8y + 6z = -31\]

Add the adjusted equations

Adding the equations, we get \[7x - 7x + 7y - 8y - 7z + 6z = 35 - 31\]which simplifies to \[-y - z = 4\]We have ended up with an identical equation as in Step 2. This means that all three planes represented by the equations intersect at a line, and therefore have an infinite number of solutions.

Key concepts

System of Equations

A system of equations is a group of two or more equations that share common variables. These systems are typically used to determine the values of the variables that satisfy every equation simultaneously. In this context, we'd be working to solve three equations with variables 'x', 'y', and 'z'.
This particular problem proposes the use of the elimination method, where the goal is to strategically cancel out one of the variables, making it easier to solve for the remaining variables.

• In the given exercise, we start by targeting the variable 'x'.
• By adjusting the equations with multiplication, we altered their structure so that adding them would cancel out 'x'.

This is often the first step in simplifying a system of equations: to bring them down to two variables from three.

Infinite Solutions

In solving a system of equations, there are three possible outcomes:

• One solution - a single point where all equations meet.
• No solution - equations that never intersect.
• Infinite solutions - a more complex result where multiple conditions lead to numerous points of intersection.

The exercise on hand ends with infinite solutions. Here's why:

• Upon eliminating the variable 'x', the resultant equations ended up being identical implying redundancy.
• This redundancy suggests the three planes of the equations don't intersect at a single point but form a line.

More intuitively, when you solve and find dependent equations - like getting the same equation more than once - it means they overlap significantly, creating infinite points that satisfy the system.

Linear Algebra

Linear algebra is a branch of mathematics focused on vector spaces and linear mappings between these spaces. It is the backbone of solving systems of equations. When we talk about systems of equations, they're usually linear, meaning they graph as straight lines (or planes, in three-dimensional cases).
Key functionalities:

• Understanding dimensionality: Plane vs. space.
• Provides tools for solving equations systematically and efficiently.
• Utilizes concepts such as matrices and vectors for more complex systems.

In this exercise, understanding linear algebra concepts is pivotal. By recognizing how equations translate to planes intersecting at lines or a single point, we gain insights into possible solutions. Linear algebra thus gives us both flexibility in methods like elimination and insight into outcomes like infinite solutions.