Problem 120 - Intermediate Algebra

Question

Solve: \(\quad 0.75(x-5)-\frac{4}{5}=\frac{1}{6}(3 x+1)+3.2\).

Step-by-Step Solution

Answer

The solution is approximately \(x \approx 31.67\).

Step 1: Distribute and Simplify

Start by distributing the terms in the equation. Distribute the 0.75 across the parentheses: \[0.75(x-5) - \frac{4}{5} = \frac{1}{6}(3x+1) + 3.2\]This becomes:\[0.75x - 3.75 - \frac{4}{5} = \frac{1}{6} imes 3x + \frac{1}{6} + 3.2\]Now simplify each side of the equation. Convert fractions to decimals for consistency: \[0.75x - 3.75 - 0.8 = 0.5x + 0.1667 + 3.2\]

Step 2: Combine Like Terms

Combine the constant terms on both sides of the equation:\[0.75x - 4.55 = 0.5x + 3.3667\]Here, -3.75 and -0.8 combine to -4.55 on the left side, and 0.1667 plus 3.2 combine to 3.3667 on the right side.

Step 3: Isolate the Variable

To isolate the variable, move all terms involving \(x\) to one side and constant terms to the other side. Subtract \(0.5x\) from both sides:\[0.75x - 0.5x = 3.3667 + 4.55\]This simplifies to:\[0.25x = 7.9167\]

Step 4: Solve for x

Solve for \(x\) by dividing both sides by 0.25:\[x = \frac{7.9167}{0.25}\]This gives:\[x = 31.6668\]Thus, \(x \approx 31.67\) when rounded to two decimal places.

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions that represent lines. These equations can generally be solved by performing operations that simplify and balance the equation on both sides. The primary goal is to isolate the variable, such as \(x\), on one side to find its value. A typical linear equation looks like this: \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
To solve:

• Combine like terms to simplify.
• Use addition or subtraction to isolate the term with the variable.
• Divide or multiply to solve for the variable.

By following these steps methodically, you can reliably find the value of the unknown variable present in the equation.

Distributive Property

The distributive property is a key tool in algebra that helps in simplifying expressions. It states that \(a(b + c) = ab + ac\). This means you multiply the term outside the parentheses by each term inside the parentheses. This property helps to break down more complex expressions into simpler ones.
Consider the expression as seen in the original exercise: \(0.75(x - 5)\). Utilizing the distributive property, you would calculate:

• \(0.75 \times x = 0.75x\)
• \(0.75 \times (-5) = -3.75\)

This makes the expression easier to handle and further moves you toward isolating the variable.

Fractions to Decimals Conversion

Converting fractions to decimals is a helpful step, especially when all terms in an equation need to be consistent for easy manipulation. This process involves dividing the numerator by the denominator.

• For the fraction \(\frac{4}{5}\), when divided, gives \(0.8\).
• For \(\frac{1}{6}\), it becomes approximately \(0.1667\).

By converting, equations that involve both fractions and decimals are more manageable. It also allows calculations within the equations to be performed smoothly. Always ensure that the decimal representation is precise enough for the required solution accuracy.

Combining Like Terms

Combining like terms involves simplifying an equation by summing constants and similar variables. Terms with the same variables (like \(x\) terms) or constants (numbers) can be algebraically combined.

• In the problem, terms \(-3.75\) and \(-0.8\) combine to give \(-4.55\).
• On the other side, \(0.1667\) and \(3.2\) sum to \(3.3667\).

This step helps to further simplify the equation, making an easier path to isolate the variable and solve the rest of the equation.