Problem 45 - Algebra and Trigonometry

Question

Solve the system of linear equations. $$ \left\\{\begin{array}{l}{x-y+6 z=8} \\ {x \quad+\quad z=5} \\ {x+3 y-14 z=-4}\end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions, dependent on the value of \( z \).

Step 1: Write the System of Equations

First, let's write down the given system of linear equations for clarity: Equation 1: \( x - y + 6z = 8 \)Equation 2: \( x + z = 5 \)Equation 3: \( x + 3y - 14z = -4 \)

Step 2: Express x in terms of z

From Equation 2: \( x + z = 5 \), we can solve for \( x \):\[ x = 5 - z \]

Step 3: Substitute x in Equations 1 and 3

Substitute \( x = 5 - z \) into Equations 1 and 3:For Equation 1: \[ (5 - z) - y + 6z = 8 \] Simplify to: \[ -y + 5z = 3 \] (We'll call this Equation 4.)For Equation 3: \[ (5 - z) + 3y - 14z = -4 \] Simplify to: \[ 3y - 15z = -9 \] (We'll call this Equation 5.)

Step 4: Solve for y from Equation 5

From Equation 5: \( 3y - 15z = -9 \), solve for \( y \):\[ 3y = 15z - 9 \] \[ y = 5z - 3 \]

Step 5: Substitute y in Equation 4

Substitute \( y = 5z - 3 \) into Equation 4:\[ - (5z - 3) + 5z = 3 \] Simplify:\[ -5z + 3 + 5z = 3 \] \[ 3 = 3 \]This is always true, indicating the system has infinitely many solutions.

Key Concepts

Solving Linear Equations

Solving a system of linear equations involves finding values for the variables that satisfy all the given equations at the same time. In a system composed of linear equations, each equation represents a straight line. When solving the system, we're looking for the point or points where these lines intersect. There are several methods to solve systems of linear equations, including graphing, substitution, and elimination. These methods can help us determine if there is one solution, no solution, or infinitely many solutions.

In our exercise, we have a system of three equations: \(x - y + 6z = 8\), \(x + z = 5\), and \(x + 3y - 14z = -4\). To begin solving, we express one of the variables in terms of the others, which simplifies the process of finding the solution.

Infinitely Many Solutions

When a system of equations has infinitely many solutions, it means that there is not just one set of variable values that satisfy all the equations. Instead, there is a whole range of such values. This typically occurs when the equations are equivalent in some way or when they represent the same plane or line in a geometric sense.

In our problem, after substituting and simplifying, we reached a trivial equality \(3 = 3\), indicating that no matter what value \(z\) takes, the system holds true. This tells us that the lines or planes represented by these equations overlap or coincide completely, thus leading to an infinite number of solutions. So, for each value of \(z\), there is a corresponding \(x\) and \(y\) that solve the entire set of equations.

Substitution Method

The substitution method is a powerful algebraic technique used to solve systems of linear equations. It involves solving one of the equations for one variable and then substituting this expression into the other equations. This simplification can help isolate one of the variables, making it easier to find solutions for the system.

In this exercise, we first solved Equation 2 for \(x\): \(x = 5 - z\). Then, we substituted this expression into Equations 1 and 3 to get new equations (Equation 4 and 5, respectively).

This approach allowed us to eventually simplify the system into an equation that was always true, evidencing the infinitely many solutions. Substitution is particularly handy when the system is not too complex and when it's feasible to easily express one variable in terms of others without cumbersome calculations.