Problem 10 - Precalculus Mathematics for Calculus

Question

Use back-substitution to solve the triangular system. $$\left\\{\begin{aligned} x-2 y+3 z &=10 \\ 2 y-z &=2 \\ 3 z &=12 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \( x = 4 \), \( y = 3 \), and \( z = 4 \).

Step 1: Solve for the last variable

Start with the simplest equation, which usually involves only one variable. In this system, the equation is \( 3z = 12 \). Solve for \( z \) by dividing both sides by 3: \( z = \frac{12}{3} = 4 \).

Step 2: Substitute to solve for the second variable

Use the value of \( z \) obtained from Step 1 and substitute it into the second equation. The equation is \( 2y - z = 2 \). Substitute \( z = 4 \) to get \( 2y - 4 = 2 \). Add 4 to both sides to isolate the term with \( y \): \( 2y = 6 \). Divide by 2 to solve for \( y \): \( y = 3 \).

Step 3: Substitute to solve for the first variable

Now substitute the values of \( y \) and \( z \) into the first equation. The equation is \( x - 2y + 3z = 10 \). Replace \( y = 3 \) and \( z = 4 \) to get \( x - 2(3) + 3(4) = 10 \), which simplifies to \( x - 6 + 12 = 10 \). Simplify further to \( x + 6 = 10 \). Subtract 6 from both sides to find \( x \): \( x = 4 \).

Step 4: Confirm the solution

Verify the solution by substituting \( x = 4 \), \( y = 3 \), and \( z = 4 \) back into the original equations. Check that all equations hold true: \( 4 - 2(3) + 3(4) = 10 \), \( 2(3) - 4 = 2 \), and \( 3(4) = 12 \). These computations confirm the correctness of the solutions.

Key Concepts

triangular system

A triangular system of equations is crucial in simplifying and solving complex problems. This system is characterized by its staircase or triangular arrangement of equations, where each subsequent equation involves one less variable than the previous one.
This format makes it much easier to solve the equations sequentially. The system is typically described in two forms: upper triangular form and lower triangular form. In an upper triangular system, the equations are arranged such that the top-left to the bottom-right diagonal contains the coefficients, with all entries below the diagonal being zero.
The key benefit of a triangular system is the ability to use back-substitution for solving, which we'll discuss more in detail. It transforms a series of simultaneous equations into a form that is straightforward to solve, equation by equation, starting with the simplest one.

solving equations

Solving equations within a triangular system typically employs a method called back-substitution. This involves starting with the simplest equation, usually involving a single variable, then working backwards through the system.

• Start with the simplest equation that typically appears at the bottom of the triangular system.
• Solve for the variable in this equation.
• Once the value is determined, substitute this value back into the previous equations to find the remaining variables.

Back-substitution leverages the simplicity of triangular systems. By solving the easiest equation first, it reduces complexity as you work upwards. This method reduces errors and provides a structured approach to finding the solution.
In our example, we started with solving for \( z \) using the simplest equation \( 3z = 12 \), then substituted \( z = 4 \) into the next equation to solve for \( y \), and finally solved for \( x \) using the values of \( y \) and \( z \).

step-by-step solution

Taking a step-by-step approach to solving triangular systems ensures accuracy and clarity. Here's how to apply this to a problem effectively:

• Step 1: Identify and solve the simplest equation first. Usually, the equation at the bottom of the triangular system involves only one variable.
• Step 2: Use the value from Step 1, substituting it into the next equation. This reduces unknowns, making it simpler to solve for the next variable.
• Step 3: Continue substituting known values into the remaining equations. This approach simplifies complex equations into manageable steps.
• Step 4: Verify the solution by substituting the values of all variables into the original equations. Ensure each equation is satisfied, confirming the solution's accuracy.

By following a clearly defined step-by-step method, solving triangular systems becomes straightforward. This procedure promotes systematic thinking and minimizes errors in complex calculations.