Problem 26 - 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. $$\left\\{ \begin{aligned} x + y + z & = 5 \\ x - 2 y + 4 z & = 13 \\ 3 y + 4 z & = 13 \end{aligned} \right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 0, y = 3, z = 2\).

Step 1: Rewrite the System

Reorder the system of equations such that equations 1 and 2 become \(x + y + z = 5\) and \(x - 2y + 4z = 13\), on subtracting equation 1 from equation 2 we get the third equation as \( -3y + 3z = 8 \).

Step 2: Solve for two variables

Now we have two equations with two variables: \(y + z = 5\) {from step 1} and \(3y + 4z = 13\). Multiply the first equation by 3 to align it with the second equation then subtract the two equations: \(3y + 3z = 15\) and \(3y + 4z = 13\). That gives us \(z = 2\). With the value of z, we can now substitute it into the first equation \(y + 2 = 5\) to find that \(y = 3\).

Step 3: Solve for x

Now substitute the values of y and z into the original first equation. \(x + 3 + 2 = 5\). That gives \(x = 0\).

Step 4: Verify the Solution

Substitute the solution \(x = 0, y = 3, z = 2\) into each of the original equations to verify if the left-hand side equals the right-hand side in all cases. They do, so the solution is correct.

Key Concepts

Algebraic Methods for Solving Systems

Solving systems of linear equations is a foundational skill in algebra. They're sets of equations with multiple variables that have a common solution. The goal is to find the values of the variables that satifsy all equations simultaneously. Some common algebraic methods for solving these systems include the substitution method, elimination method, and using matrices and determinants.

For systems that can be easily rearranged to isolate one variable, the substitution method is often the quickest approach. The elimination method involves adding or subtracting equations to eliminate one variable, making it simpler to solve. For larger systems, matrices can be used alongside operations such as row reduction to find solutions more systematically. No matter the method chosen, the process demands careful manipulation of equations to maintain the equality balanced on both sides.

Substitution Method

The substitution method is an algebraic technique used when one equation in a system can be solved for one variable in terms of the others. This isolated variable can then 'substitute' into the other equations. Here’s how it typically works:

• Isolate one variable in one of the equations.
• Substitute the expression for this isolated variable into the other equation(s).
• Solve the new equation for one variable, then use this value to find others.

For instance, in our exercise, we first found a relationship between y and z, and then plugged the value of z back into that relationship to find y. Afterward, y and z values were substituted back into the original equation to find x. This method is particularly useful for systems where rearranging terms is straightforward, and it can make what looks like a complex problem much simpler.

Verification of Solutions

After finding the values of the variables, it’s critical to verify that they indeed solve the entire system of equations. We do this by plugging the solutions back into each original equation and checking if the equations hold true.

To verify, follow these steps:

• Substitute the solution set into each original equation.
• Ensure that the left-hand side of the equation equals the right-hand side.

If the solution set satisfies all equations, then it is indeed the correct solution to the system. In our exercise, the verification process demonstrated that the values of x, y, and z obtained through the substitution method are correct, reaffirming their accuracy. Verification assures us that no computational mistakes were made and gives confidence in the solutions found.