Problem 14 - Algebra and Trigonometry

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(5 x-3=4\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{7}{5}\).

Step 1: Add 3 to Both Sides

The given equation is \(5x - 3 = 4\). To simplify, we need to isolate the variable term \(5x\). First, add 3 to both sides to cancel out the \(-3\) on the left side: \(5x - 3 + 3 = 4 + 3\) This simplifies to: \(5x = 7\).

Step 2: Divide Both Sides by 5

Now that we have \(5x = 7\), the next step is to isolate \(x\) by dividing both sides by 5. This gives: \(\frac{5x}{5} = \frac{7}{5}\) The equation simplifies to: \(x = \frac{7}{5}\).

Key Concepts

Linear Equations

Linear equations form the backbone of algebra. They are equations in which the highest power of the variable is one. For example, in the equation \(5x - 3 = 4\), the variable \(x\) is raised to the first power. Linear equations can be represented in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. These equations are called 'linear' because, if plotted on a graph, they produce a straight line.
Linear equations are pervasive in math because they model real-world situations where relationships remain consistent. They are used to find unknown quantities when relationships dictate fixed proportions between quantities. This is what makes mastering linear equations such an invaluable skill for students.

Isolation of Variables

Isolation of variables is a crucial part of solving linear equations. The goal is to get the variable by itself on one side of the equation. This means manipulating the equation through algebraic operations like addition, subtraction, multiplication, and division.
In the original problem \(5x - 3 = 4\), we started by adding 3 to both sides of the equation to eliminate the -3. This step is necessary to ensure that the only term left with \(x\) is not hindered by other constants. Think of it as simplifying the path to uncover the value of \(x\).

• First, remove any constants from the variable term using addition or subtraction.
• Second, handle any coefficients attached to the variable using multiplication or division.

This orderly approach is what allows us to isolate the variable smoothly.

Step-by-Step Solutions

Step-by-step solutions break the problem-solving process into manageable parts. This approach makes solving equations less intimidating and more systematic. A step-by-step method allows for clearer thought and understanding, eliminating potential errors.
Let's examine the solution steps again:

• **Step 1: Add 3 to Both Sides** - This was done to move the constant to the same side as the other number, simplifying \(5x - 3 = 4\) to \(5x = 7\).
• **Step 2: Divide Both Sides by 5** - To isolate \(x\), divide everything by 5, resulting in \(x = \frac{7}{5}\).

Each step focuses on a particular task, ensuring that no detail is overlooked. Such a methodical approach is essential not only for solving linear equations but also for tackling other types of algebraic manipulations.