Problem 82 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a calculator to solve each equation. $$x-7.0463=-9.2714$$

Step-by-Step Solution

Answer

The value of 'x' in the given equation is \(-2.2251\).

Step 1: Understand the Equation

You are given the equation \(x - 7.0463 = -9.2714\). This is a simple linear equation which you are tasked to solve for 'x'.

Step 2: Isolate 'x'

The main goal here is to isolate 'x'. Doing so involves adding \(7.0463\) on both sides of the equation to cancel out the \(-7.0463\) on the left side. Hence, the equation transforms to \(x = -9.2714 + 7.0463\).

Step 3: Calculate the result

After the equation is simplified as described, it is necessary to perform the addition. Since it is indicated to use a calculator, perform the operation \(-9.2714 + 7.0463\) to get a precise decimal answer.

Key Concepts

Using a Calculator

Calculators can simplify solving equations immensely, especially when you're dealing with complex decimals or large numbers. They allow you to quickly perform operations such as addition, subtraction, multiplication, or division, which saves time and reduces the risk of manual calculation errors.

For the given equation, you need to calculate \(-9.2714 + 7.0463\). Rather than performing this manually, a calculator streamlines the process:

• First, turn on your calculator and ensure you're in the standard or basic mode.
• Enter the value \(-9.2714\).
• Press the addition (+) button.
• Enter the second number, \(7.0463\).
• Finally, press the equal (=) button to get the result.

This approach provides an accurate and efficient way to solve arithmetic operations, giving you the decimal result which is crucial for further analysis in equations.

Isolating Variables

When solving linear equations, isolating the variable is generally your primary objective. This means you need to get the unknown (in this case, \(x\)) by itself on one side of the equation.

The equation is \(x - 7.0463 = -9.2714\). To isolate \(x\), you need to perform the inverse operation to eliminate the constant on the same side as \(x\):

• Add \(7.0463\) to both sides of the equation to nullify the negative term. This is done because addition is the reverse process of subtraction.
• Apply this to both sides, transforming it into \(x = -9.2714 + 7.0463\).

Through these steps, you have successfully isolated \(x\), preparing the equation for a simple computation.

Decimal Operations

Working with decimal numbers in equations can sometimes be daunting due to their precision and the arithmetic involved. However, understanding how to handle decimals is essential for solving equations accurately.

In the given equation, you deal with decimals: \(x - 7.0463 = -9.2714\). Here's a simple way to manage them:

• Align decimal points when adding or subtracting numbers. This ensures each place value is correctly calculated.
• Make sure your calculator is set to an appropriate precision level if it allows, such as two or more decimal places, to ensure the correct amount of decimal digits is preserved.
• When adding \(-9.2714 + 7.0463\), maintain consistency by handling each digit post-decimal properly so that the result reflects accurate arithmetic.

These practices help maintain precision and accuracy, making you more confident in your solutions.