Problem 11 - Precalculus: Mathematics for Calculus

Question

Use back-substitution to solve the triangular system. \(\left\\{\begin{aligned} 2 x-y+6 z &=5 \\ y+4 z &=0 \\\\-2 z &=1 \end{aligned}\right.\)

Step-by-Step Solution

Answer

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

Step 1: Solve the Last Equation

Start by solving the last equation, \(-2z = 1\) \, for \(z\). Divide both sides by -2: \[z = -\frac{1}{2}.\]

Step 2: Substitute z into the Second Equation

Substitute the value of \(z\), which is \(-\frac{1}{2}\), into the second equation, \(y + 4z = 0 \), to solve for \(y\): \\(y + 4(-\frac{1}{2}) = 0\).\Simplify: \\(y - 2 = 0\).\Thus, \y = 2.

Step 3: Substitute y and z into the First Equation

Substitute \(y = 2\) and \(z = -\frac{1}{2}\) into the first equation, \(2x - y + 6z = 5\), to solve for \(x\): \\(2x - 2 + 6(-\frac{1}{2}) = 5\).\Simplifying: \\(2x - 2 - 3 = 5\),\which gives \\(2x - 5 = 5\).\Add 5 to both sides: \\(2x = 10\).\Divide by 2: \\(x = 5\).

Key Concepts

Triangular Systems

A triangular system is a set of linear equations where each equation has fewer variables than the previous one.This structure makes these systems easier to solve, step-by-step. Generally, triangular systems are encountered in two forms:

• Upper Triangular Systems: Equations are arranged in a descending order by the number of variables.
• Lower Triangular Systems: Equations are arranged in an ascending order by the number of variables.

In our specific exercise, we have an upper triangular system:

• The first equation has three variables: \(2x - y + 6z = 5\).
• The second equation has two variables: \(y + 4z = 0\).
• The third equation has one variable: \(-2z = 1\).

By solving the equations from the bottom up, one variable at a time, we convert the steps into a straightforward back-substitution process.First, solve for the last variable, then move upwards until all variables are determined.

Linear Equations

Linear equations are mathematical statements of equality between algebraic expressions.They represent straight lines when graphed on a coordinate plane. In general, a linear equation in two variables can be written in the form:\[ ax + by = c \]where:

• \(a\) and \(b\) are coefficients, possibly zero.
• \(x\) and \(y\) are variables.
• \(c\) is a constant.

When dealing with linear systems, the goal is to find the values of variables that satisfy all given equations simultaneously.In our triangular system exercise, we've encountered a series of linear equations involving three variables: \(x\), \(y\), and \(z\).These equations intersect at a single point, providing a unique solution.Solving such systems requires methods like substitution or elimination, but in this well-ordered form, back-substitution is used.

Precalculus

Precalculus serves as the foundation for calculus and involves concepts from algebra and trigonometry. It prepares students by developing skills to solve complex mathematical problems. Triangular systems of linear equations often surface in precalculus to build algebraic and analytical skills.

• Understanding functions and equation manipulation.
• Building skills in recognizing and solving systems of equations.
• Prepares students for calculus through understanding solutions involving rates of change and slopes.

The success in these topics can significantly impact the ease with which students transition to calculus. In this exercise, tackling a triangular system exemplifies the practical use of algebra to solve real-world problems, a key aim of precalculus.

Algebra

Algebra involves manipulating symbols and equations to solve problems. It is a branch of mathematics dealing with expressions and the rules for manipulating these expressions. Within algebra, solving systems of equations is essential for decoding real-life scenarios into mathematical solutions.

• Defines how we express unknown quantities in equations.
• Involves simplifying and rearranging equations to find solutions.
• Provides tools for handling complex and useful mathematical models.

In the triangular system exercise, algebraic methods are employed to manage multiple equations efficiently. Each equation in the system is sequentially reduced using algebraic operations until the values of all variables are known. Understanding algebra thus becomes powerful for unraveling structured problems such as this triangular system, demonstrating algebra's significant role in problem-solving.