Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-17,\) solve each equation or inequality. Each solution is an integer. $$ 5 x+4=39 $$

Step-by-Step Solution

Answer

The solution is \(x = 7\).

Step 1: Isolate the Variable Term

The given equation is \(5x + 4 = 39\). To isolate the term containing the variable \(x\), we first need to remove the constant on the left side. Do this by subtracting 4 from both sides:\[5x + 4 - 4 = 39 - 4\]The equation simplifies to:\[5x = 35\]

Step 2: Solve for the Variable

Now that we have \(5x = 35\), the next step is to solve for \(x\) by dividing both sides of the equation by 5:\[\frac{5x}{5} = \frac{35}{5}\]This simplifies to:\[x = 7\]

Key Concepts

Understanding Equations

Equations are fundamental concepts in algebra that express the equality between two expressions. In its simplest form, an equation consists of two sides with an equal sign between them. Each side can contain numbers, variables, or a combination of both.
These mathematical statements are powerful tools for modeling and solving real-world problems.
The aim is to find the value of the variable that makes the equation true.
The given equation is a linear equation, which is characterized by constants and variables where the variables are to the first power only. Linear equations such as this one, play a crucial role in algebra because they are easy to solve and form the basis for more complex mathematical concepts.
To solve an equation, we follow specific operations like addition, subtraction, multiplication, and division. These operations are used to isolate the variable on one side of the equation, leading us to the solution. Remember, whatever operation is performed on one side of an equation, must be performed on the other side to maintain balance.

The Meaning of Integer Solutions

An integer is a whole number, and when a problem requires integer solutions, it means the answer must be a whole number that might be positive, negative, or zero.
In the context of the equation provided, the instructive part is that the solution, which is the value of the variable, must be an integer.
Integers are fundamental in mathematics as they simplify computation and are easier to interpret in most practical situations. When dealing with linear equations, as we are in this exercise, the outcome often results in an integer.
Here, after performing the steps to isolate the variable, we find that the integer solution is 7. A clear understanding of what it means to have an integer solution helps reinforce why this specific solution completes the task.

Process of Solving Equations

Solving equations involves several critical steps to determine the value of the variable. This process is methodical and follows specific algebraic rules. Let’s break it down using the equation from the example:

• Isolate the Variable: Start by moving terms not involving the variable to the other side of the equation. In the example, we subtract 4 from both sides to achieve this balance.
• Simplify the Equation: Once terms are moved, simplify each side of the equation, so it becomes clearer. Here, we reduced the problem to a simpler form, 5x = 35.
• Solve for the Variable: Use division or multiplication to isolate the variable. We divide both sides by 5 to solve for x, resulting in x = 7.

Each step brings you closer to finding the solution, and each decision is based on ensuring the equation stays balanced.
The key is to perform the same operation on both sides of the equation, which maintains equivalence and leads directly to the answer.