Problem 53 - Elementary Algebra Within Reach

Question

Solve the equation. Round your answer to two decimal places. $$0.234 x+1=2.805$$

Step-by-Step Solution

Answer

The solution to the equation is \(x \approx 7.71\).

Step 1: Subtract one from both sides

Subtract 1 from both sides of the equation to isolate the term with \(x\). This results in \(0.234x = 2.805 - 1\), which simplifies to \(0.234x = 1.805\).

Step 2: Divide by the coefficient of \(x\)

Divide both sides of the equation by the coefficient of \(x\), which is 0.234. This results in \(x = \frac{1.805}{0.234}\).

Step 3: Solve for \(x\)

Performing the division gives \(x \approx 7.71\).

Key Concepts

Rounding Decimals

When solving equations, rounding decimals is often necessary to simplify the solution. For example, in the given exercise, we were asked to round the final answer to two decimal places. This means we should display only the two digits after the decimal point. Rounding helps in making numbers easier to read and work with, especially when dealing with long decimal places.

• If the third decimal digit is 5 or more, we increase the second decimal place by one.
• If the third decimal digit is less than 5, we leave the second decimal place as is.

By applying these rules to the solution of the equation, where we found that dividing 1.805 by 0.234 approximately equals 7.711, we take only two decimal places. Therefore, it becomes 7.71, which is a more concise and easy-to-manage number.

Isolation of Variables

To solve a linear equation like the one given, the first step involves isolating the variable. This is done by rearranging the equation so that the variable you're solving for, in this case, \(x\), stands alone on one side of the equation.
For the equation \(0.234x + 1 = 2.805\), we start by getting rid of any constants added or subtracted from the variable term. Here, we subtract 1 from both sides, leading to \(0.234x = 1.805\).

• Subtract or add numbers from both sides to leave the variable and its coefficient.
• This ensures that the unit containing the variable is separated from other numbers, simplifying further steps in solving the equation.

Isolation is crucial because it cleans up the equation, making it easier to see the relationship between the variable and the constants.

Division in Equations

Division is often used in the final step of solving linear equations, especially when a coefficient is attached to a variable. In our exercise, once we isolated the term \(0.234x\), we needed to solve for \(x\) itself by dividing both sides by its coefficient, which was 0.234.
Here's how it looks:

• Divide each side of the equation by the coefficient of \(x\) to get \(x\) alone.
• This operation ensures that \(x\) is isolated completely, resulting in the value of \(x\).

Applying this to the problem, dividing 1.805 by 0.234 gives \(x \approx 7.71\) after rounding. Remembering these division steps in equations is vital for turning expressions into solutions.