Problem 51 - Elementary Algebra Within Reach

Question

Solve the equation. Round your answer to two decimal places. $$1.2 x-4.3=1.7$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5.00\).

Step 1: Identify the Problem

We are trying to solve the equation \(1.2x - 4.3 = 1.7\) for the variable \(x\).

Step 2: Add 4.3 to Both Sides.

We have to isolate \(x\). We start by adding \(4.3\) to both sides of the equation in order to remove \(-4.3\) from the left side of the equation. This results in \(1.2x = 1.7 + 4.3\). That simplifies to \(1.2x = 6.0\)

Step 3: Solve for x.

Now to isolate \(x\), we divide each side of the equation by \(1.2\). This will provide the solution for \(x\) as \(x = \frac{6.0}{1.2}\).

Key Concepts

Solving Linear Equations

Linear equations are equations of the first degree, which means they involve variables raised only to the power of one. The main aim when solving linear equations is to find the value of the variable, such as \(x\), that makes the equation true. In the equation \(1.2x - 4.3 = 1.7\), we need to determine the value of \(x\).

The general steps for solving linear equations include:

• Eliminate any constants on the side containing the variable by performing inverse operations, such as addition or subtraction to both sides of the equation.
• Once the variables are isolated on one side, divide or multiply to solve for the variable.
• Check your solution by substituting the variable back into the original equation to see if both sides equal.

Being familiar with these steps makes it easier to solve linear equations and find the value of the variable involved.

Rounding Decimals

Rounding decimals is the process of reducing the number of digits after the decimal point for ease of use or clarity, while keeping the number's overall value close to what it was before rounding. When solving equations like \(x = \frac{6.0}{1.2}\), you may end up with a result that has many decimal places. This might not be practical or necessary, so rounding helps simplify the number to a form that is easier to use or understand.

For instance, the solution to the equation \(x = 5\), when calculated exactly, can be a repeating decimal. However, if the instruction is to round to two decimal places, you would simplify your solution accordingly.

To round to two decimal places:

• Locate the third decimal place.
• If this digit is 5 or greater, increase the digit in the second decimal place by one.
• If it is less than 5, leave the second decimal place digit as it is.

Rounding helps provide clear and manageable answers without losing significant accuracy.

Isolation of Variables

When solving equations, isolating the variable is key to finding its value. This process involves rearranging the equation so that the variable stands alone on one side. In the given equation \(1.2x - 4.3 = 1.7\), we seek to isolate \(x\).

To begin isolating \(x\), we need to eliminate any numbers added to or subtracted from \(x\). We start by adding \(4.3\) to both sides to remove the \(-4.3\), giving us \(1.2x = 6.0\).

Next, we divide each side by \(1.2\) in order to leave \(x\) on its own, achieving the solution \(x = \frac{6.0}{1.2}\).

Properly isolating the variable ensures that the solution is both accurate and clearly shows the value of the variable in question.