Problem 12 - College Algebra with Modeling and Visualization

Question

Determine whether the equation is linear or nonlinear by trying to write it in the form ax \(+b=0\) $$ 2(x-3)=4-5 x $$

Step-by-Step Solution

Answer

The equation is linear.

Step 1: Distribute Terms

Apply the distributive property to eliminate parentheses on the left side of the equation: \[ 2(x-3) = 2 \cdot x - 2 \cdot 3 = 2x - 6 \] The equation becomes: \[ 2x - 6 = 4 - 5x \]

Step 2: Move All Terms to One Side

Add \(5x\) to both sides to move all terms to one side of the equation:\[ 2x + 5x - 6 = 4 \] Which simplifies to: \[ 7x - 6 = 4 \]

Step 3: Isolate Variable x

Add 6 to both sides to isolate terms with \(x\):\[ 7x = 10 \]

Step 4: Check the Form

In the step above, we ended with the equation \(7x = 10\), which can be rewritten in the standard linear form \(ax + b = 0\) by writing \(7x - 10 = 0\). Since \(a = 7\) is not equal to 0, this equation represents a linear equation.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps you to simplify and manipulate equations. It allows you to multiply a single term across a set of terms inside parentheses. For example, in the expression \(2(x-3)\), the distributive property allows you to distribute the \(2\) to both \(x\) and \(-3\). Here's how it works:

• Multiply \(2\) by \(x\) to get \(2x\).
• Multiply \(2\) by \(-3\) to get \(-6\).

Therefore, \(2(x-3)\) becomes \(2x - 6\). This simplification is essential to clearing parentheses and making equations easier to work with.
This technique is extensively used in algebra to simplify expressions before solving equations.
Once distributed, the expression transitions into a more workable form, easing the path to isolating variables.

Isolation of Variables

The isolation of variables is a process where you manipulate an equation to get the variable you are trying to solve for on one side of the equation by itself. In essence, it's making one side of the equation look something like \(x = ...\). In the given exercise, after applying the distributive property, the equation is \(2x - 6 = 4 - 5x\). The goal is to have all terms containing the variable \(x\) on one side. Here's how that's accomplished:

• Add \(5x\) to both sides to combine similar terms and eliminate \(x\) from the right side.
• This results in \(7x - 6 = 4\).

Next, isolate \(x\) by moving any constants to the opposite side by addition or subtraction. In this case, you would add \(6\) to both sides:

• This gives us \(7x = 10\).

Thus, the variable is isolated, allowing you to proceed further in solving it. This method is crucial in rearranging equations to get the needed variable alone.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. Once the variable is isolated, as shown in the previous section, the equation typically simplifies into an instance where arithmetic can solve for the variable.For example, with our equation \(7x = 10\), you will solve for \(x\) by dividing both sides of the equation by \(7\):

• \(x = \frac{10}{7}\)

Now, \(x\) is expressed as \(\frac{10}{7}\), which is a specific value solving the original equation. Converting the equation into the linear form \(ax + b = 0\) assures its linearity as every term becomes a multiple of the variable \(x\). Linear equations are distinct due to their defining forms and predictable solutions which underpin much of algebraic operations.
By expressing the equation this way, you ensure no terms remain outside the criteria of a linear function.