Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} &x-z=2\\\ &x+y=-3\\\ &y-z=1 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \((x, y, z) = (-1, -2, -3)\).

Step 1: Express x in terms of z

From the first equation, we have \( x - z = 2 \). Solve for \( x \) to get \( x = z + 2 \).

Step 2: Express y in terms of z

From the third equation, we have \( y - z = 1 \). Solve for \( y \) to get \( y = z + 1 \).

Step 3: Substitute into the second equation

Substitute the expressions for \( x \) and \( y \) from Step 1 and Step 2 into the second equation \( x + y = -3 \). This gives \((z + 2) + (z + 1) = -3 \).

Step 4: Simplify and solve for z

Simplify the equation from Step 3: \(2z + 3 = -3 \). Solve for \( z: 2z = -6 \) which gives \( z = -3 \).

Step 5: Determine x and y values

Use the value of \( z = -3 \) back in the expressions for \( x \) and \( y \). For \( x: x = z + 2 \to x = -3 + 2 = -1 \). For \( y: y = z + 1 \to y = -3 + 1 = -2 \).

Step 6: Verify the solution

Substitute \( x = -1 \), \( y = -2 \), and \( z = -3 \) back into the original equations to confirm they satisfy all equations:1. \( x - z = -1 - (-3) = 2 \)2. \( x + y = -1 + (-2) = -3 \)3. \( y - z = -2 - (-3) = 1 \).Since all are satisfied, the calculations are confirmed.

Key Concepts

System of Equations

A system of equations is essentially a set of equations with multiple variables. Solving it means finding values for the variables that satisfy all equations in the system. In our example, we have three equations involving the variables \(x\), \(y\), and \(z\):

• \(x - z = 2\)
• \(x + y = -3\)
• \(y - z = 1\)

To solve such systems analytically, we employ methods that find precise solutions, such as substitution or elimination, rather than approximations. Analytical solutions reveal whether the system is consistent (having at least one solution) and provide insight into the relationships between variables.

Dependent Equations

Dependent equations are equations within a system that are not independent of each other. They effectively describe the same condition in different forms, which can lead to an infinite number of solutions that form a line or plane.In the given system, we derive expressions for \(x\) and \(y\) in terms of \(z\):

• From \(x - z = 2\), we find \(x = z + 2\).
• From \(y - z = 1\), we find \(y = z + 1\).

After substituting these back into the second equation, it allows all equations to hold true because each variable is dependent on \(z\). The consistency of these dependencies shows how these equations describe a line of solutions in three-dimensional space.

Substitution Method

The substitution method is a strategy used to solve systems of equations where you solve one of the equations for one variable, and then substitute that expression into the other equations. This method simplifies the problem by reducing the number of variables and solving step by step.Here’s how the substitution method works in our example:1. Solve \(x - z = 2\) for \(x\), getting \(x = z + 2\).2. Solve \(y - z = 1\) for \(y\), getting \(y = z + 1\).3. Substitute these expressions into \(x + y = -3\) to find an equation solely in terms of \(z\): \((z + 2) + (z + 1) = -3\).4. This simplified equation allows us to solve for \(z = -3\).

Solution Verification

Verifying your solution is a critical step in solving systems of equations. It ensures that the proposed solution not only satisfies one equation, but is consistent with all equations in the system.To verify, we substitute the values of \(x = -1\), \(y = -2\), and \(z = -3\) back into each of the original equations:

• Substitute into \(x - z = 2\): \(-1 - (-3) = 2\), which is true.
• Substitute into \(x + y = -3\): \(-1 + (-2) = -3\), which is true.
• Substitute into \(y - z = 1\): \(-2 - (-3) = 1\), which is true.

Each substitution confirms the correctness of our solution, signifying that these values indeed satisfy all the conditions given in the system of equations.