Problem 8 - College Algebra with Modeling and Visualization

Question

Determine whether the equation is linear or nonlinear by trying to write it in the form ax \(+b=0\) $$ 100-23 x=20 x $$

Step-by-Step Solution

Answer

The equation is linear.

Step 1: Identify Terms and Rearrange

Start by identifying the terms present in the equation. The given equation is \(100 - 23x = 20x\). To rearrange into a standard form, move all terms involving \(x\) to one side of the equation and constant terms to the other side. This can be done by adding \(23x\) to both sides:\[100 = 20x + 23x\]

Step 2: Simplify the Equation

Combine like terms on the right-hand side. Since both terms involve \(x\), we combine them by adding:\[100 = 43x\]

Step 3: Rewrite in the Standard Linear Form

Next, we rewrite the equation in the standard linear form \(ax + b = 0\). Subtract \(100\) from both sides to make it equal to zero:\[43x - 100 = 0\]

Step 4: Identify the Coefficients

Compare the equation \(43x - 100 = 0\) with the standard linear form \(ax + b = 0\). Here, \(a = 43\) and \(b = -100\).

Step 5: Determine if the Equation is Linear

Since the equation can be written in the form \(ax + b = 0\), where \(a\) and \(b\) are constants, the equation is linear.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving linear equations. It involves rearranging and simplifying expressions to uncover variables or solve for unknowns. The first step is to identify each term in the equation. Consider the given equation:

• Start with identifying the terms. In the example equation, we see terms involving constants and variables: \(100 - 23x = 20x\).
• The next step is rearranging: bring similar terms together. To achieve this, add or subtract terms across the equal sign. For instance, in our example, add \(23x\) to both sides to isolate the constant term:\[100 = 20x + 23x\]

This adjustment creates an equation where like terms are grouped. Simplifying \(20x + 23x\) results in \(43x\). Algebraic manipulation helps in bringing clarity and simplifying the path to solving the equation.

Standard Form of a Linear Equation

The standard form of a linear equation is an essential concept in algebra because it provides a clear way to identify linear relationships. This format is typically expressed as \(ax + b = 0\).

• After simplifying an equation, the goal is to express it in this standard form. In the example, rearranging gives us:\[43x - 100 = 0\]
• Here, \(43x\) represents the variable term involving the coefficient of \(x\), and \(-100\) is the constant term.

Understanding this structure makes it easier to analyze and solve equations. By rewriting equations in the standard form, identifying and comparing the components becomes straightforward, allowing for efficient problem-solving.

Identifying Coefficients

Identifying coefficients is a critical step in working with linear equations. Coefficients are constants that multiply the variable. In the standard form \(ax + b = 0\),

• \(a\) represents the coefficient of the variable \(x\).
• In our example, we identified \(a = 43\) and \(b = -100\).

Recognizing these values helps quantify the linear relationship and determine equation parameters. Coefficients also assist in understanding the slope and intersection of linear graphs if applied in graphing contexts. Proper identification of coefficients simplifies analysis and provides clarity in resolving linear equations.