Question

Solve the system of equations. $$\left\\{\begin{array}{rr}2 x+3 y+z= & -4 \\ 2 x-4 y+3 z= & 18 \\ 3 x-2 y+2 z= & 9\end{array}\right.$$

Short answer

The solution to the system of equations is (x, y, z) = (1, 3, -1).

Step-by-step solution

Setting Up the Equation System

Set the given system of equations with three variables as follows: \[\begin{{align*}}2x + 3y + z & = -4 \2x - 4y + 3z & = 18 \3x - 2y + 2z & = 9 \\end{{align*}}\]

Eliminate One Variable

Take the first two equations and create a new system of equations to eliminate variable x. The new system of equations becomes:\[\begin{{align*}}7y - 2z & = 22 \-2y + z & = 5 \\end{{align*}}\]

Solve for the Remaining Variables

Now, we multiply the second equation by 2 and add it to the first, which gives us y = 3. Substituting y = 3 into the second equation gives z = -1.

Substitution into Original Equations

Substitute y = 3 and z = -1 into the original third equation, which gives x = 1.

Key concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that we aim to solve simultaneously. In the context of this exercise, we have a set of three linear equations with three unknowns: x, y, and z. To find a solution means to determine the set of values for x, y, and z that satisfy all equations at the same time.

It's analogous to finding a point where three planes intersect in three-dimensional space. In some cases, systems can have a single solution (one point of intersection), infinitely many solutions (planes coincide), or no solution at all (planes are parallel).

• Single Solution: Unique intersection point.
• Infinitely Many Solutions: Planes overlap entirely.
• No Solution: Planes never intersect.

Variable Elimination

Variable elimination, often used in systems like the one we are considering, helps us simplify the complex problem into more manageable chunks. The aim here is to eliminate one variable so that we reduce our system to a two-variable problem, which is easier to handle.

This is typically achieved by either adding or subtracting equations from each other. In our exercise, by manipulating the first two equations, we eliminate x and redirect our focus to finding the values of y and z. The goal is to continue eliminating variables until we are left with one equation with one variable, which is straightforward to solve.

Substitution Method

The substitution method can be especially powerful after elimination has been used to simplify the system. Once we've isolated a variable in one equation (for example, y in terms of z), we can substitute this expression into another equation, thus eliminating that variable from the second equation.

In our exercise, after eliminating x, we were able to find the value of y (y=3). We then 'substituted' this value into another equation to find z. The substitution method acts like locks and keys, where finding the right key (the value of one variable) can unlock the values of others.

Algebraic Solutions

Algebraic solutions encapsulate the end goal of solving systems of equations—finding the numerical values that satisfy all the equations algebraically. Once we've used methods like elimination and substitution to simplify the system, we calculate the exact values by following basic algebraic principles.

In this case, we proceeded step-by-step: first, we eliminated a variable; second, we substituted to find another variable's value; and finally, we back-substituted these known values into the original equations to find all unknowns. This provides a clear, stepwise path to the full algebraic solution of the system.