Problem 62 - Elementary Algebra Within Reach

Question

Solve the equation. Round your answer to two decimal places. $$\frac{x}{4.08}+7.2=5.14$$

Step-by-Step Solution

Answer

The solution to the equation is x = -8.37

Step 1: Identify and isolate the terms with the unknown variable

Separate the term with the unknown (x) by using subtraction operation. Therefore, \(\frac{x}{4.08} = 5.14 - 7.2\).

Step 2: Calculate the right side

Perform the subtraction on the right side of the equation to get \(\frac{x}{4.08} = -2.06\).

Step 3: Solve for unknown variable

To solve for x, multiply both sides of the equation by 4.08 giving x = -2.06 * 4.08.

Step 4: Evaluate x

Perform the multiplication to get the value of x as -8.37, when rounded off to two decimal places.

Key Concepts

Isolating the Variable

In algebra, to find the value of an unknown variable, one of the initial steps is isolating the variable. This involves manipulating the equation to have the unknown variable on one side and the known values or terms on the other. The goal is to have the variable in a form such as 'x = ...', so we can easily determine its value. In our example, we start with the equation \( \frac{x}{4.08} + 7.2 = 5.14 \).

To isolate x, we need to remove 7.2 from the left side. We do this by performing the opposite operation, which, in this case, is subtraction. We subtract 7.2 from both sides, leading to \( \frac{x}{4.08} = 5.14 - 7.2 \). We now have an equation with x on one side, which is the first step towards finding its value. It's important to always perform the same operation on both sides of the equation to maintain balance, which is fundamental in preserving the equation's integrity.

Performing Operations with Decimals

Working with decimals is common in algebra, and accuracy is key. Performing operations with decimals follows the same rules as with whole numbers, but attention to detail is a must to ensure the correct placement of the decimal point. In our exercise, we encounter a subtraction: \(5.14 - 7.2 \).

When subtracting, align the decimal points first. It can be helpful to rewrite the numbers vertically: 5.14- 7.20This alignment allows us to subtract methodically from right to left. Here, 5.14 becomes smaller than 7.20, leading to a negative result. When conducting any operation with decimals—be it addition, subtraction, multiplication, or division—carefully track the decimal point to avoid errors in the final result. In the solution, after subtracting, we get \( -2.06 \), a critical step in solving for x.

Evaluating Expressions

After simplifying the equation and isolating the variable, the next phase is evaluating expressions to find the value of the unknown. This often involves performing arithmetic operations with the remaining terms. To solve for x in the given equation \( \frac{x}{4.08} = -2.06 \), we need to get rid of the denominator, 4.08, linked to x.

Multiplying both sides of the equation by 4.08 cancels out the denominator on the left side and isolates x. The multiplication: \( -2.06 \times 4.08 \), must be performed with care to preserve the correct sign and decimal place. After the multiplication, we conclude with x = -8.37, rounded to two decimal places. Evaluating expressions is a critical step in solving equations as it directly leads to the solution. It's important to apply the correct arithmetic operations and round the answers appropriately when specified, as it's often the case in real-world scenarios where precise measurements are required.