Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation (if possible). $$5 x+3=6-2 x$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 3 / 7 \).

Step 1: Rearrange the terms involving x

We aim to get all terms involving \( x \) on one side of the equation and the constants on the other side. To achieve this, we add \( 2x \) to both sides of the equation to move \( -2x \) from the right side to the left side. Doing this results in \( 5x + 2x + 3 = 6 + 2x - 2x \). The \( 2x \) terms cancel out on the right, leaving us with \( 7x + 3 = 6 \).

Step 2: Isolate x

We still have a constant, 3, on the left side of the equation with the \( x \) terms. We can move it to the right side by subtracting 3 from both sides of the equation. This gives us \( 7x + 3 - 3 = 6 - 3 \), simplifying to \( 7x = 3 \).

Step 3: Solve for x

Finally, we solve for \( x \) by dividing the entire equation by 7, which is the coefficient of \( x \). This gives us \( x = 3 / 7 \).

Key Concepts

Rearranging Terms

When it comes to solving linear equations, rearranging terms is a foundational skill. Each side of the equation contains terms that may include the variable we're solving for, as well as constant numbers. By rearranging these terms, we aim to simplify the equation and eventually isolate the variable.
Let's illustrate this with our example: \(5x + 3 = 6 - 2x\). Notice that terms with \(x\) are located on both the left and right sides of the equation. Our goal in this first step is to concentrate all \(x\) terms on one side and simplify the constants on the other.
Here are the simple steps to rearrange terms:

• Identify the terms containing the variable.
• Move variable terms to one side by adding or subtracting them from both sides. In our case, we add \(2x\) to both sides to get rid of the \(-2x\) on the right.
• Combine like terms.

This rearrangement resulted in \(7x + 3 = 6\). Thus, all \(x\) terms are now on one side, making it easier to proceed to the next step.

Isolating the Variable

Once we've managed to concentrate all \(x\) terms on one side, our next task is to isolate the variable. The reason behind isolating the variable is to figure out the specific value of \(x\) that makes the equation true.
In our current example, after rearranging, we have \(7x + 3 = 6\). To isolate \(x\), we need to handle the constants and any coefficients attached to the \(x\) term.
Follow these steps:

• First, remove the constant from the side of the equation containing the variable by adding or subtracting it. Here, subtract \(3\) from both sides to clear it from the left side.
• Reorganize your equation to reflect this change, leading to \(7x = 3\).

With the constant removed, we can focus on solving for \(x\) by removing any coefficients.

Basic Algebra

Basic algebra involves the operations and techniques needed to manipulate equations and find solutions for unknown variables. It is essential to understand the arithmetic rules that govern these operations, such as addition, subtraction, multiplication, and division.
When solving equations like \(7x = 3\), the basic algebra skill needed is to perform operations that will yield the value of \(x\). The term \("solve for \(x\)"\) means doing whatever operation necessary, based on the arithmetic rules, to get \(x\) by itself on one side of the equation.
In this example, we divide both sides by \(7\) to isolate \(x\):

• Divide both sides by the coefficient of \(x\) (which is \(7\) in this equation) to solve for \(x\).
• This division yields \(x = \frac{3}{7}\).

Having a solid grasp of basic algebra makes solving linear equations a straightforward process, as you can see from the methodical steps we've taken to isolate \(x\) and find its value.