Problem 1 - Elementary Algebra

Question

Solve each of the equations. $$x-0.36=0.75$$

Step-by-Step Solution

Answer

The value of \(x\) is 1.11.

Step 1: Understand the Goal

The goal is to find the value of \(x\) that will satisfy the equation \(x - 0.36 = 0.75\).

Step 2: Isolate the Variable

To solve for \(x\), the variable should be isolated on one side of the equation. This can be achieved by adding \(0.36\) to both sides of the equation.

Step 3: Perform the Calculation

Add \(0.36\) to both sides of the equation: \[x - 0.36 + 0.36 = 0.75 + 0.36\] The equation simplifies to: \[x = 1.11\]

Step 4: Conclusion

The value of \(x\) that satisfies the equation is \(1.11\). You have isolated \(x\) by performing the necessary arithmetic operation.

Key Concepts

Isolation of Variables

When solving linear equations, one of the key steps is focusing on isolating the variable of interest, such as \( x \). This means rearranging the equation in a manner that \( x \) stands alone on one side of the equation. Achieving this helps us to directly pinpoint what \( x \) equals.

Consider our exercise: \( x - 0.36 = 0.75 \). Our task is to zero in on \( x \), so that by the end of the process, it sits isolated on one side of the equation.

To do this, you might often need to employ operations like addition or subtraction, to cancel terms paired with the variable. These operations are performed on both sides of the equation. For example, if our equation subtracts \( 0.36 \) from \( x \), the reverse operation is to add \( 0.36 \), which we apply equally to both sides. This ensures that the balance of the equation is maintained, leading us to a clear expression of \( x \).

Arithmetic Operations

Arithmetic operations are the backbone of solving equations. These operations include addition, subtraction, multiplication, and division, which are used to systematically simplify and solve for variables.

Let's take a closer look at our example: after isolating \( x \), we come to the point where we need to perform specific operations to find the value of \( x \). In the equation \( x - 0.36 = 0.75 \), adding \( 0.36 \) on both sides serves to eliminate the \( -0.36 \) beside \( x \).

This is how it works step by step:

• Add \( 0.36 \) to \( x - 0.36 \). This results in \( x \), because \( -0.36 + 0.36 = 0 \).
• Next, add \( 0.36 \) to the right side of the equation, \( 0.75 \), resulting in \( 1.11 \).

These calculated steps confirm that the arithmetic operations were applied correctly, yielding the solution \( x = 1.11 \). As you can see, arithmetic ensures that every step in equation-solving dismantles the puzzle, piece by piece.

Equation Solving Steps

Solving an equation systematically is crucial for arriving at an accurate solution. Following a sequence of steps helps in understanding and solving the problem clearly.

Here's a breakdown of how to approach the equation \( x - 0.36 = 0.75 \):

• **Step 1:** Understand the ProblemIdentifying your goal is the first step. In this case, it's finding the value of \( x \) that balances the equation.
• **Step 2:** Isolate the VariableShift focus to \( x \) by eliminating any terms attached to it. We did this by adding \( 0.36 \) to both sides of the equation.
• **Step 3:** Perform Arithmetic OperationsThis involves the actual calculations that solve the equation. After isolating \( x \), perform the addition: \( 0.75 + 0.36 \), which gives \( 1.11 \).
• **Step 4:** Conclude with ConfidenceReaching a solution, confirm by plugging \( x = 1.11 \) back into the original equation to see if it holds true. If it does, you've successfully solved the equation.

Each of these steps ensures a structured approach that minimizes errors and enhances understanding. Following them helps you grasp the why and how of each operation used in solving linear equations.