Problem 55 - Elementary Algebra Within Reach

Question

Solve the equation. Round your answer to two decimal places. $$3+0.03 x=5$$

Step-by-Step Solution

Answer

The solution to the equation is \(x \approx 66.67\)

Step 1: Subtract 3 from both sides

To gather all the x terms on one side and numbers on the other side, subtract 3 from both sides of the equation. This gives \(0.03x = 5 - 3\), which simplifies to \(0.03x = 2\).

Step 2: Isolate x

In order to isolate x, divide both sides of the equation by 0.03. This gives \(x = \frac{2}{0.03}\)

Step 3: Find the numerical value of x

Perform the division to find the numerical value of x, rounding to two decimal places. \(x \approx 66.67\)

Key Concepts

Understanding Algebra

Algebra is an essential branch of mathematics that helps us describe relationships using symbols and letters. At its core, it involves operations similar to arithmetic but uses variables to stand for unknown values.💡 A common activity in algebra is solving equations, which is what we see in this exercise. Here, the equation given is linear because its variable, represented by \(x\), is to the power of one.

When tackling algebraic equations:

• You aim to isolate the unknown variable (in this instance, \(x\)) on one side of the equation.
• This involves using inverse operations to simplify and solve the equation step by step.

In our exercise, we start by eliminating constant numbers from the side with the variable. This is a crucial skill in algebra, as it sets the foundation for isolating \(x\). By using the rule of inverse operations, like subtracting to cancel adding, we moved closer to finding the value of \(x\).

Remember, algebra is not just about numbers. It's an approach and set of tools you apply to solve real-world problems.

Equation Solving Strategies

Equation solving is like a puzzle—you have to find the piece that fits perfectly. The aim is to find the value of the variable that makes the equation true. For our exercise, the task is to simplify and find the value of \(x\). To do this, we follow several steps.

Initially, you need to move the constants away from the variable. For example, from the equation \(3 + 0.03x = 5\), we subtract 3 from both sides to simplify it, resulting in \(0.03x = 2\).
This makes the structure cleaner and shows a clear pathway to isolate the variable.

Once the variable term is isolated, the next strategy involves getting the variable alone. This requires dividing the equation by the coefficient of \(x\), which is \(0.03\) in this scenario. So, we divide both sides, simplifying it to \(x = \frac{2}{0.03}\).

This strategy is consistent for linear equations and aids in managing more complex equations by breaking them into simpler chunks.

Using Decimal Approximation

Decimal approximation is an important skill, especially when dealing with real numbers. It helps to represent numbers in a simpler form without losing significant accuracy.

In this exercise, after dividing \(2\) by \(0.03\), we find \(x = 66.6666...\). Instead of writing the repeating decimal, you approximate it to a finite number of decimal places. You round to two decimal places if specified, providing clarity and neatness to your result.

• This means you report the number as \(x \approx 66.67\).
• Rounding involves looking at the digit beyond where you wish to round to decide if you should increase the last digit shown.

Decimal approximation is practical in everyday math, finance, and science, where exact values are often impractical. It's about balancing precision with ease of use, making numbers manageable to the everyday learner.