Problem 54 - Elementary Algebra Within Reach

Question

Solve the equation. Round your answer to two decimal places. $$2.75 x-3.13=5.12$$

Step-by-Step Solution

Answer

x = 3.00

Step 1 - Add 3.13 to both sides of the equation

This step results in:\[ 2.75x = 5.12 + 3.13 \]

Step 2 - Simplify the right-hand side of the equation

The right-hand side simplifies to 8.25, resulting in:\[ 2.75x = 8.25 \]

Step 3 - Divide by 2.75

Dividing both sides by 2.75 isolates \(x\), so:\[ x = 8.25 / 2.75 \]

Key Concepts

Rounding Decimals

When solving equations and needing a precise answer, rounding decimals becomes crucial. In this exercise, you'll often find yourself with numbers that have long decimal points, which can be cumbersome to work with. Fortunately, rounding helps make these numbers more manageable and easier to interpret.
To round a decimal number, you first need to decide how many decimal places you want. In this exercise, we're asked for two decimal places.

• Look at the third decimal place: if it's 5 or higher, round the second decimal place up.
• If it's lower, round down.

For instance, if our final calculation results in a number like 3.0072, you would look at the third decimal place, which is 7. Since it's higher than 5, you'd round 0.0072 to 3.01. This approach keeps your answer both precise enough for practical use and easier to understand.

Equation Simplification

Before we can get to the answer, simplifying the equation is a critical step. At first glance, equations might seem complex, but breaking them down bit by bit makes them manageable.
Simplifying equations involves performing mathematical operations to bring similar terms together, or to make the equation easier to solve.

• First, address any constants on either side of the equation, like adding or subtracting numbers.
• Then, combine like terms to streamline the equation.

In our example: \[2.75x - 3.13 = 5.12 \]we start by eliminating the -3.13 by adding 3.13 to both sides. This simplification step transforms our equation into \[2.75x = 8.25.\]This strategy allows us to zero in on the key component of the equation more easily.

Isolation of Variable

The objective of solving any equation is to determine the value of the variable, which, in our example, is \(x\). Achieving this requires isolating the variable, which means getting the variable alone on one side of the equation.
Here's how it works:

• First, perform operations that will remove numbers or coefficients directly connected to the variable.
• Use inverse operations, like division if the variable is multiplied, or subtraction if it is added.

In the given problem: \[2.75x = 8.25,\]we divide both sides by 2.75, which is the coefficient of \(x\). By doing so, we successfully isolate \(x\) and find that\[x = 8.25 / 2.75,\]which simplifies to \(x = 3\). By isolating the variable, we effectively solve the equation.