Problem 31 - Precalculus with Limits

Question

Solving a System of Linear Equations In Exercises \(25 - 46\) , solve the system of linear equations and check any solutions algebraically. $$ \left\\{ \begin{array} { l } { 3 x - 5 y + 5 z = 1 } \\ { 5 x - 2 y + 3 z = 0 } \\ { 7 x - y + 3 z = 0 } \end{array} \right. $$

Step-by-Step Solution

Answer

The solution to the system is \(x = 1\), \(y = -1\), and \(z = \frac{5}{2}\).

Step 1: Simplify the equations

Firstly, rewrite the system in a simplified form by arranging the coefficients in an orderly manner. It can be written as:\[ \begin{align*} 3x - 5y + 5z &= 1 \ 5x - 2y + 3z &= 0 \ 7x - y + 3z &= 0 \end{align*} \]

Step 2: Solve the system by substitution or elimination

It is possible to solve the system either by substitution or elimination. Since none of the given equations can be easily solved for a single variable, we should try elimination. Multiply the first equation by 2 and the second one by 5, afterwards subtract the second from the first. This will eliminate \(y\). Do the same with the third equation by multiplying it by 2 and the first equation by 7, then subtract the first from the third to remove \(y\). This leads to: \[\begin{align*}11x + 15z &= 2\7x + 17z &= 7\end{align*}\]

Step 3: Solve for one variable

Next, multiply the first equation by 7 and the second one by 11, and then subtract the first from the second to remove \(x\). This yields \(2z = 5\), hence \(z = \frac{5}{2}\). Substitute \(z = \frac{5}{2}\) back into the first two new equations obtained in the last step to get \(x=1\), and into the original system to get \( y = -1\).

Step 4: Check the solution

Finally, make sure to verify this solution by substituting the obtained values \((1, -1, \frac{5}{2})\) back into the original system. If the values satisfy every equation, then the solution is valid.

Key Concepts

Elimination Method

The elimination method is a way to solve a system of linear equations by strategically eliminating one variable at a time. This approach involves combining pairs of equations to gradually reduce the number of variables until each equation contains only one. In the given problem, the equations include variables \(x\), \(y\), and \(z\). Because the equations are complex, elimination is chosen as the preferred method.

First, we focus on eliminating \(y\). By multiplying equations and then subtracting, we aim to cancel out \(y\) completely:

• Equation 1: Multiply by 2
• Equation 2: Multiply by 5
• Subtract (Equation 2 from Equation 1): Resulting in an equation with only \(x\) and \(z\)

By repeating similar steps for other combinations, such as pairing the second and third equations, we achieve two new equations that are simpler and have only two variables. This step-by-step removal of variables is what defines the elimination method. Ultimately, the goal is to continue the process until you isolate one variable, making it possible to solve for the others.

Substitution Method

The substitution method is another technique to solve systems of linear equations. Although it wasn't directly used in this particular problem, understanding it can provide additional insights. The essence of substitution is to solve one of the equations for a single variable and substitute that expression into the other equations.

Here's how substitution generally works:

• Solve one equation for one of the variables (e.g., solve for \(x\) in terms of \(y\) and \(z\))
• Substitute this expression into the other equations to reduce the number of variables
• Continue simplifying until you obtain a single equation with a single variable, which you can solve directly

Substitution is particularly helpful when one variable is already isolated or can be easily isolated, making the substitution process straightforward. While the original system in the example wasn't ideal for substitution, it can be a powerful method in other equations where the relationships are simpler.

Algebraic Verification

Once a solution is found for a system of linear equations, checking its correctness is crucial. Algebraic verification is the process of substituting the found values back into the original equations to ensure they satisfy all conditions.

This final step includes:

• Substituting the values of \(x = 1\), \(y = -1\), and \(z = \frac{5}{2}\) back into all original equations
• Checking that each equation is balanced, meaning both sides of the equals sign have the same value

Verification not only confirms the solution's accuracy but also clarifies whether any computational or algebraic mistakes were made. Through this reconfirmation process, students ensure that their solutions are both valid and thorough, providing a strong foundation for understanding these methods.