Problem 11 - College Algebra with Modeling and Visualization

Question

Determine whether the equation is linear or nonlinear by trying to write it in the form ax \(+b=0\) $$ 7 x-5=3(x-8) $$

Step-by-Step Solution

Answer

The equation is linear.

Step 1: Simplify the Equation

Start by expanding the right side of the equation. The equation is given as \(7x - 5 = 3(x - 8)\). First, distribute the 3 across the terms in the parentheses: \(3(x - 8) = 3x - 24\). Thus, the equation becomes \(7x - 5 = 3x - 24\).

Step 2: Move Variable Terms to One Side

Next, isolate the terms containing \(x\) on one side. Subtract \(3x\) from both sides of the equation: \(7x - 3x - 5 = -24\). This simplifies to \(4x - 5 = -24\).

Step 3: Solve for x

Add 5 to both sides to further simplify: \(4x = -24 + 5\). Adding gives \(4x = -19\). Divide both sides by 4: \(x = \frac{-19}{4}\). The final form of the equation can be arranged as \(4x + 19 = 0\).

Step 4: Determine Linearity

The equation is now in the form \(ax + b = 0\) where \(a = 4\) and \(b = 19\). There are no exponents on \(x\) other than 1, meaning the equation is linear.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is about finding the unknown or putting real-life variables into equations and then solving them. When you think of algebra, think about equations, symbols like "x" and "y," and finding the values they represent.
In the given exercise, algebra is at the heart of what we're doing. We start with an equation and use algebraic methods to simplify and solve it. Understanding algebra helps you move terms around and apply rules like the distributive property, which is used when expanding expressions such as

• Distributive Property: Distribute numbers over a sum, like "3\( (x-8) = 3x - 24 \)," applying multiplication across terms in parentheses.
• Simplification: Combine like terms, for instance, moving all terms involving "x" onto one side of the equation.

By mastering algebra, you can solve more complex equations and understand the structure of mathematical problems better.

Equation Solving

Equation solving is the process of finding the value or values of the variable that make a given equation true. It's about figuring out what values replace the unknowns to make both sides of the equation equal. In equations like the one from the exercise "7x - 5 = 3(x - 8)," the goal is to find what "x" equals by making logical moves.

Steps in Solving Linear Equations

• Expand and Simplify: Like in the exercise, first expand the equation "7x - 5 = 3x - 24" by using distributive property to simplify terms.
• Rearrange Terms: Move terms involving the variable "x" to one side by using addition or subtraction. This results in a simpler equation like "4x - 5 = -24."
• Isolate the Variable: Adding 5 to each side gives "4x = -19," then divide by the coefficient of "x," which is in this case 4, to find the solution: \(x = \frac{-19}{4}\).

Each step is essential in arriving at the mathematical solution. Solving equations teaches you to think critically and analytically, seeing the logic in mathematical operations.

Mathematical Modeling

Mathematical modeling involves using mathematical language and equations to represent and solve real-world problems. This is where math meets the world! You transform real-life situations into mathematical equations, which you can analyze to make predictions or decisions. In our exercise, although it seems straightforward, we're essentially modeling a situation where some conditions need to be balanced or fulfilled. By modeling, we can:

• Use Linear Equations: Represent relationships that are directly proportional and can be solved to predict specific outcomes.
• Simplify Complex Problems: By breaking down a problem and using equations to represent it, complexities are reduced, and solutions become more visible.

Modeling builds a bridge between abstract mathematics and practical applications, allowing you to see how mathematical equations can predict and solve problems in science, engineering, economics, and everyday life. Taking equations from textbooks and applying them to real-world is the essence of mathematical modeling.