Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{c} 5 x-3 y+2 z=3 \\ 2 x+4 y-z=7 \\ x-11 y+4 z=3 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the given system of linear equations is: x = 3, y = -\frac{5}{17} and z = \frac{18}{17}.

Step 1: Solving for x

To solve for x, eliminate y and z by subtracting the second equation from the first and the third from the second respectively. Doing this gives: \[3x = 9 \] So, \[x = 3\]

Step 2: Solving for y

Now substitute x = 3 into the first and second equations to eliminate x, then subtract the second from the first to get: \[17y = -5\]So, \[y = -\frac{5}{17}\]

Step 3: Solving for z

Substitute x = 3 and y = -\frac{5}{17} into any of the original equations to get z. Using the third equation gives: \[z = \frac{18}{17} \]

Step 4: Check the solution

Finally, substitute x = 3, y = -\frac{5}{17}, z = \frac{18}{17} back into the original equations to verify if the left hand side equals to the right hand side for all equations.

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, and the linear transformations between them. It is foundational for understanding how multiple equations can interact and overlap in a shared space. Linear equations, like the ones in our exercise, involve variables raised to the first power. These equations can represent lines, planes, or hyperplanes in multi-dimensional space. With systems of linear equations, each equation is a constraint, and the solution is a point (or set of points) that satisfies all given constraints. In this exercise, we are presented with three linear equations with three variables: x, y, and z. The task is to find the values for these variables that make all the equations true at the same time.

Solution Verification

Verifying the solution through algebraic means ensures that each obtained solution satisfies the original set of equations. This step is crucial to confirm the accuracy of computed values. In our problem, after calculating the values for x, y, and z using the substitution method, we plug these numbers back into the original equations. This involves substituting x = 3, y = -\(\frac{5}{17}\), and z = \(\frac{18}{17}\) into each equation to check equality of both sides. If all three equations are satisfied, then the solution is verified.

• Confirms both sides of the equation are equal
• Ensures the solution is mathematically sound
• Validates problem-solving accuracy

This process is a safeguard against arithmetic or logical errors made during calculations.

Substitution Method

The substitution method involves solving one of the equations for one variable and using this expression to substitute into other equations. This reduces the number of variables, simplifying the equations to make solving feasible. In this exercise, we solved for x first, getting x = 3, then substituted into the next equations to solve for y, ultimately arriving at y = -\(\frac{5}{17}\). Finally, we used known values of x and y to calculate z using any of the original equations. This method is useful:

• For systems where one equation is easily solvable for one variable
• When other methods are too complex or time-consuming
• To maintain a straightforward pathway in problem solving

Problem Solving Steps

Breaking down the problem-solving process into clear steps aids in organizing the approach to tackle systems of linear equations. Here, the structured steps involved were: 1. **Isolate one variable:** By manipulating the equations to solve for a single variable, complexity is reduced. 2. **Substitute back:** Use the found value to reduce remaining equations, simplifying through substitution. 3. **Solve step-by-step:** With reduced equations, solve systematically for remaining variables. 4. **Verify:** Always check solutions against original equations. Each step logically builds on the previous, ensuring no detail is overlooked. By navigating the problem through these steps, the chances of errors are minimized, and understanding is maximized.