Problem 57 - Elementary Algebra

Question

Solve each of the following equations and express the solutions in decimal form. Your calculator might be of some help. $$7 x-0.39=0.03$$

Step-by-Step Solution

Answer

The solution is \(x = 0.06\).

Step 1: Isolate the Variable

To solve for \(x\), we need to start by isolating the variable on one side of the equation. We'll do this by eliminating the constant term on the left side of the equation. The original equation is: \[ 7x - 0.39 = 0.03 \]Add \(0.39\) to both sides to get rid of the constant term on the left:\[ 7x = 0.03 + 0.39 \]\[ 7x = 0.42 \]

Step 2: Solve for x

Now that we have the equation \(7x = 0.42\), to solve for \(x\), divide both sides of the equation by \(7\): \[ x = \frac{0.42}{7} \]

Step 3: Calculate the Result

Use a calculator to divide \(0.42\) by \(7\): \[ x = 0.06 \].Therefore, the solution to the equation is \(x = 0.06\).

Key Concepts

Isolating the Variable

When solving linear equations, the main goal is to find the value of the unknown variable, here represented as \(x\). The process is often called "isolating the variable," meaning we want \(x\) by itself on one side of the equation.

In our exercise, we started with the equation \[7x - 0.39 = 0.03\]The term \(-0.39\) is a constant that is subtracted from \(7x\). To isolate \(x\), the first step is to remove this constant from the left side. You achieve this by adding \(0.39\) to both sides of the equation. This is the opposite operation of subtraction, effectively canceling out the \(-0.39\) on the side with \(x\).

The equation becomes:\[7x = 0.03 + 0.39\] This simplifies to:\[7x = 0.42\]

By moving \(0.39\) to the other side, you have isolated \(7x\). This method of moving terms around by performing opposite operations is a critical step when you solve linear equations.

Decimal Solutions

Working with decimals requires attention to detail, especially in calculations. Decimal solutions are simply results expressed in decimal form, offering precision in mathematics.

Once you have isolated \(7x\) in the equation:\[7x = 0.42\] You need to solve for \(x\). Since \(x\) is multiplied by 7, you'll perform the opposite operation: division.

This means dividing both sides of the equation by 7:\[x = \frac{0.42}{7}\]

Performing this division gives you:\[x = 0.06\] In many cases, calculators can help, but understanding the concept ensures you're capable of handling any decimal number efficiently.

Keep in mind:

• Always line up decimals vertically when adding or subtracting.
• When dividing, the count of decimal places in the divisor matters.

Using a Calculator for Equations

Calculators are wonderful tools for ensuring accuracy, especially when working with decimal numbers! For many students, learning when and how to use a calculator can simplify complex problems.

In our exercise, once we isolated \(x\) to\[x = \frac{0.42}{7}\]dividing decimals by hand can sometimes lead to errors, so using a calculator here is wise. Simply input \(0.42\), divide by 7, and the calculator shows the result: \(0.06\).

Keep a few points in mind:

• Always double-check your decimal placement in entries and results.
• Make sure your calculator is set to the correct mode, often "standard mode" suffices.

Using calculators efficiently provides not just speed, but confidence in the solutions you provide, ensuring you grasp both the process and the accuracy needed in math problems.