Problem 38 - Algebra 1

Question

Solve the equation. Round the result to the nearest hundredth. $$ 6.1(3.1+2.5 x)=15.3 x-3.9 $$

Step-by-Step Solution

Answer

The solution \(x\) is -5.77.

Step 1: Expand the parentheses

The equation is \(6.1(3.1+2.5x) = 15.3x-3.9\). The first task is to expand the parentheses: \(6.1 \times 3.1 + 6.1 \times 2.5x = 15.3x-3.9\) which simplifies to \(18.91 + 15.25x = 15.3x - 3.9\) .

Step 2: Group like terms

Next, group like terms together. This gives: \(3.95x = -22.81\) . In order to isolate \(x\), you need to divide both sides by 3.95. This gives: \(x = -22.81 / 3.95\).

Step 3: Calculate division and round

Perform the division: \(x = -5.77\). The task demands that we round the result to the nearest hundredth, so there is no need for additional work here.

Key Concepts

Solving Linear Equations

Solving linear equations is a foundational skill in algebra. These equations are called "linear" because they represent straight lines when graphed. A linear equation is in the form: \( ax + b = c \), where \( a, b, \) and \( c \) are constants, and \( x \) is the variable we need to solve for. The goal is to find the value of \( x \) that makes the equation true.

• First, simplify both sides of the equation independently.
• Next, use addition or subtraction to group all terms containing the variable on one side, and constant terms on the other.
• Finally, use multiplication or division to isolate the variable and solve for \( x \).

In the given exercise, you must perform arithmetic operations and rearrange terms to isolate the variable. This involves moving terms and variables across the equal sign and ensuring that you perform the same operation on both sides. Remember, the balance of the equation must always be maintained.

Equation Expansion

Equation expansion involves removing parentheses by distributing the factors outside the parenthesis across each term inside. This is also known as the distributive property and is fundamental in simplifying equations.
To apply equation expansion:

• Multiply each term inside the parenthesis by the factor outside.
• Simplify the resulting expression by combining like terms, if possible.

In the original problem, the expression \( 6.1(3.1+2.5x) \) is expanded to become \( 6.1 \times 3.1 + 6.1 \times 2.5x \). This leads to a simplified version: \( 18.91 + 15.25x \). By expanding, you simplify complex parts of an equation, making it easier to solve in subsequent steps.

Rounding Numbers

Rounding numbers is crucial when exact answers are not practical or when you want to make a number easier to understand. In mathematics, rounding to the nearest hundredth is a common practice, especially in problems involving decimals.
Here's how you round a number to the nearest hundredth:

• Identify the hundredth place, which is the second digit after the decimal point.
• Look at the third digit after the decimal point to determine rounding. If it is 5 or greater, increase the hundredth position by 1; if it is less than 5, leave it as is.

In our solution, the division result \( x = -5.772 \) is rounded to \( x = -5.77 \) because the third decimal place is 2, which is less than 5. Rounding is particularly useful in making results more presentable and easier to interpret, especially when precision beyond a certain decimal is unnecessary.