Problem 28 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ 6(3-2 x)=1-(2 x-1) $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{8}{5} \) or \( x = 1.6 \) after checking.

Step 1: Simplify Both Sides

First, let's distribute the number inside the parentheses on both sides of the equation.For the left side: \[ 6(3 - 2x) = 18 - 12x \] For the right side: \[ 1 - (2x - 1) = 1 - 2x + 1 = 2 - 2x \]

Step 2: Set the Equation Equal

Now we have simplified both sides which gives us: \[ 18 - 12x = 2 - 2x \]

Step 3: Move All Terms Involving x to One Side

Add \( 12x \) to both sides to eliminate \( x \) on the left side:\[ 18 = 2 + 10x \]

Step 4: Isolate the Variable Term

Subtract 2 from both sides to solve for the term involving \( x \):\[ 18 - 2 = 10x \]\[ 16 = 10x \]

Step 5: Solve for x

Now, divide both sides by 10 to solve for \( x \):\[ x = \frac{16}{10} \]Simplify the fraction:\[ x = \frac{8}{5} \] or \( x = 1.6 \)

Step 6: Check the Solution

Substitute \( x = \frac{8}{5} \) into the original equation to check:\[ 6(3 - 2 \times \frac{8}{5}) = 1 - (2 \times \frac{8}{5} - 1) \]Calculate each side:Left side:\[ 6(3 - \frac{16}{5}) = 6(\frac{15}{5} - \frac{16}{5}) = 6(-\frac{1}{5}) = -\frac{6}{5} \]Right side:\[ 1 - (\frac{16}{5} - 1) = 1 - (\frac{16}{5} - \frac{5}{5}) = 1 - \frac{11}{5} = \frac{5}{5} - \frac{11}{5} = -\frac{6}{5} \]Both sides are equal. Therefore, \( x = \frac{8}{5} \) is correct.

Key Concepts

Distributive Property

The distributive property is a crucial concept in algebra that lets you multiply a term outside the parentheses with each term inside. It's like "sharing" multiplication over addition or subtraction. In this exercise, we deal with expressions on both sides of an equation. For example:

• On the left: we have \( 6(3 - 2x) \). By applying the distributive property, we multiply 6 with 3, and 6 with \(-2x\). This gives us: \( 18 - 12x \).
• On the right: we have \( 1 - (2x - 1) \). Applying the distributive property here means multiplying -1 by both terms inside the parentheses. This simplifies to \( 1 - 2x + 1 \), which further simplifies to \( 2 - 2x \).

By applying the distributive property, we have simplified the equation, making it easier to solve. This step helps us prepare for isolating the variable in the next stage.

Variable Isolation

Variable isolation is the process of getting a variable, like \( x \), by itself on one side of the equation. We use arithmetic operations to "move" other terms to the opposite side. Here's how it works in our problem:

• First, we simplify the equation to \( 18 - 12x = 2 - 2x \).
• Next, to isolate \( x \), we perform operations to remove terms involving \( x \) from one side. Add \( 12x \) to both sides to eliminate \(-12x\) on the left, resulting in \( 18 = 2 + 10x \).
• Then, subtract 2 from both sides to remove the constant on the right: \( 18 - 2 = 10x \), simplifying to \( 16 = 10x \).
• Finally, divide both sides by 10 to solve for \( x \): \( x = \frac{16}{10} \), which simplifies to \( x = \frac{8}{5} \) or \( x = 1.6 \).

Successfully isolating the variable is crucial as it tells us the value of \( x \) that satisfies the equation.

Checking Solutions

After finding the solution, it's important to verify that it's correct by plugging the value back into the original equation. This ensures both sides of the equation match, confirming the solution's correctness. Let's break it down:

• Substitute \( x = \frac{8}{5} \) into the original equation: \( 6(3 - 2 \times \frac{8}{5}) = 1 - (2 \times \frac{8}{5} - 1) \).
• Calculate the left side: \( 6(3 - \frac{16}{5}) = 6(\frac{15}{5} - \frac{16}{5}) = 6(-\frac{1}{5}) = -\frac{6}{5} \).
• Compute the right side: \( 1 - (\frac{16}{5} - 1) = 1 - (\frac{16}{5} - \frac{5}{5}) = 1 - \frac{11}{5} = \frac{5}{5} - \frac{11}{5} = -\frac{6}{5} \).

Both sides of the equation equal \(-\frac{6}{5}\), confirming that \( x = \frac{8}{5} \) is the correct solution. Checking solutions is a vital step to ensure our calculations are accurate.