Problem 31 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$3.2+x=7.5$$

Step-by-Step Solution

Answer

The solution to the equation \(3.2+x=7.5\) is \(x=4.3\).

Step 1: Subtract 3.2 from both sides

To isolate \(x\), subtract 3.2 from each side of the equation. \(3.2 + x - 3.2 = 7.5 - 3.2\) simplifies to \(x = 7.5 - 3.2\).

Step 2: Compute the value of \(x\)

Perform the subtraction operation in Step 1 to determine the value of \(x\): \(x = 7.5 - 3.2 = 4.3\).

Step 3: Check the solution

Substitute \(x = 4.3\) back into the original equation \(3.2 + x = 7.5\). This yields \(3.2 + 4.3 = 7.5\), which is true. So, the solution \(x = 4.3\) is correct.

Key Concepts

Solving Equations

When we talk about solving equations, we refer to the process of finding the value of the unknown variable that makes the equation true. Equations can involve various arithmetic operations such as addition, subtraction, multiplication, and division. In this exercise, the overall goal is to find out what value of \( x \) satisfies the equation \( 3.2 + x = 7.5 \).

To solve this particular equation, you need to ensure that both sides of the equation remain balanced. The addition property of equality is a fundamental concept that aids in this process. It states that if you add or subtract the same number from both sides of an equation, the equation remains true. Understanding this concept is crucial for effectively solving equations.

Follow these general steps to solve an equation:

• Identify the operation involving the variable on one side of the equation.
• Use inverse operations to isolate the variable.
• Check the solution by substituting the value back into the original equation.

These steps provide a framework for tackling a variety of equations in mathematics.

Isolation of Variables

Isolation of variables is an essential step in solving equations as it involves manipulating the equation to get the variable by itself on one side of the equation. This makes it easier to determine the value of the variable. In our exercise, the equation is \( 3.2 + x = 7.5 \). To isolate \( x \), we need to remove the constant 3.2 that is added to it.

An effective way to do this is by using the inverse operation. Since 3.2 is added to \( x \), we subtract 3.2 from both sides to maintain the balance of the equation. This step transforms the equation into \( x = 7.5 - 3.2 \).

• The concept of isolation ensures that the variable stands alone, making it easier to identify its value.
• Performing the same operation on both sides retains the equation's balance, keeping it valid.

Once the variable is isolated, you can compute its exact value with ease, as shown in the solution.

Arithmetic Operations

Arithmetic operations are the building blocks of solving equations. They include addition, subtraction, multiplication, and division. In this problem, subtraction is the key operation used to isolate the variable. The original equation is \( 3.2 + x = 7.5 \), and by subtracting 3.2 from both sides, we are applying the arithmetic operation appropriately.

The result of performing \( 7.5 - 3.2 \) gives us \( x = 4.3 \). It's essential to have a good understanding of how these operations work, as they are used not only to isolate variables but also to verify solutions. When you perform each operation correctly, you ensure that the solution derived is accurate.

Here are some quick reminders:

• Addition and subtraction are inverse operations. They undo each other.
• Understanding when to add or subtract can simplify solving linear equations.

These concepts create a solid foundation for solving more complex mathematical problems as you continue learning.