Problem 39 - Elementary Algebra

Question

Solve. $$ 5 x+7=32 $$

Step-by-Step Solution

Answer

\(x = 5\)

Step 1: Simplify the Equation

To isolate the term with the variable \(x\), we need to remove the constant on its side. Currently, the equation is \(5x + 7 = 32\). To eliminate the \(+7\), subtract 7 from both sides. This gives us \(5x + 7 - 7 = 32 - 7\), which simplifies to \(5x = 25\).

Step 2: Solve for x

Now that the equation is \(5x = 25\), we need \(x\) by itself. To do this, divide both sides by 5 to solve for \(x\): \(\frac{5x}{5} = \frac{25}{5}\). This simplifies to \(x = 5\).

Key Concepts

Equation Solving

When tackling algebraic problems, one of the main skills students need to master is equation solving. An equation is a mathematical statement where two expressions are set equal to each other. In the given exercise, the equation is \(5x + 7 = 32\). Solving an equation involves finding a value for the variable that makes this equation true.

To achieve this, we often perform 'balancing operations'. These operations involve performing the same mathematical actions on both sides of the equation to maintain equality. For example, if a number is added to one side, it must also be subtracted from the other. This is crucial because it preserves the balance of the equation while working toward isolating the variable. By following these steps, you can solve any simple linear equation.

Linear Equations

Linear equations are equations involving a single variable raised to the power of one, without any higher powers, roots, or complex operations. They form straight lines when graphed and are foundational in algebra. An example of a linear equation is \(5x + 7 = 32\), where \(x\) is the variable.

• The general form of a linear equation in one variable is \(ax + b = c\).
• The goal is to solve for \(x\) by performing appropriate operations that simplify the expression.
• These equations often appear in problem-solving scenarios and have numerous practical applications, such as determining rates or assessing profit and loss.

Understanding linear equations is crucial because they lead to more advanced algebraic concepts. They also help in developing logical and sequential thinking skills, vital across various fields of study.

Variable Isolation

Variable isolation is the process of rearranging an equation such that the variable of interest is by itself on one side of the equation. This allows us to easily identify its value. In our example, the task was to isolate \(x\) in the equation \(5x + 7 = 32\).

To isolate the variable, you should:

• Identify any constants attached to the variable (like the \(+7\) in our equation) and perform inverse operations to remove them. Here, we subtracted 7 from both sides.
• Next, modify the equation so that \(x\) stands alone. This typically involves dividing or multiplying both sides by the coefficient of \(x\), which in our case was 5.

By isolating the variable, you simplify your equation into a form where the value of the variable can be directly seen. Practicing this skill is key to conquering more complex algebraic challenges with ease.