Problem 1 - Algebra Form and Function

Question

Solve the equations in Exercises \(1-14\). \(0.5 x-3=11\)

Step-by-Step Solution

Answer

Answer: \(x = 28\)

Step 1: Add 3 to both sides of the equation

To remove the constant term from the left side of the equation, we will add 3 to both sides: \(0.5x - 3 + 3 = 11 + 3\)

Step 2: Simplify the equation

Now, we simplify the equation by performing the addition: \(0.5x = 14\)

Step 3: Divide both sides by 0.5

To isolate the variable \(x\), we need to divide both sides of the equation by the coefficient of \(x\), which is 0.5: \(\frac{0.5x}{0.5} = \frac{14}{0.5}\)

Step 4: Solve for x

Now, we can simply divide to solve for \(x\): \(x = 28\) Therefore, the solution to the equation \(0.5x - 3 = 11\) is \(x = 28\).

Key Concepts

Solving Equations

Understanding how to solve equations is a fundamental skill in algebra. Equations are mathematical sentences that express the equality of two expressions. When solving equations, our goal is to find the unknown value, known as the variable, that makes the equation true.

In the given exercise, the equation is a linear one: \(0.5x - 3 = 11\). "Linear" means that the variable \(x\) is raised only to the power of one. Solving linear equations involves a few basic steps:

• Manipulate the equation to combine like terms.
• Use inverse operations to simplify and eliminate parts of the equation that do not contain the variable.
• Perform similar operations on both sides of the equation to maintain equality.

This structured approach ensures that we systematically isolate the variable and find its value.

Mastering these techniques will create a foundation for more complex mathematical problem-solving skills.

Algebraic Manipulation

Algebraic manipulation is all about rearranging and simplifying expressions to make equations easier to work with. In our exercise, algebraic manipulation helps in transitioning from \(0.5x - 3 = 11\) to the solution step by step. One of the first things we do is add or subtract numbers to both sides.

To start, the constant \(-3\) is added to both sides of the equation: \(0.5x - 3 + 3 = 11 + 3\). The reason this works is because it uses the addition property of equality—which tells us that adding the same number to both sides of the equation keeps the equation balanced.

After removing the constant part, the equation is simplified to \(0.5x = 14\). Simplification forms a crucial part of algebraic manipulation, allowing us to see the equation in its simplest form. Making the equation as simple as possible prepares it for the next steps towards solving for the variable.

Variable Isolation

Variable isolation is the process of getting the variable alone on one side of the equation. This is a key step since it reveals the value of \(x\), the unknown we are solving for.

Once the equation was simplified to \(0.5x = 14\), the next step was to isolate \(x\). This involved dividing both sides of the equation by \(0.5\)—the coefficient of \(x\). In essence, we used the division property of equality: \(\frac{0.5x}{0.5} = \frac{14}{0.5}\).

This property allows us to divide both sides of the equation by the same non-zero number without changing the equality. Carrying out this division resulted in finding \(x = 28\).

• Isolate the variable using inverse operations.
• Ensure the solutions make the original equation true.

This step concludes the process and reveals the needed variable, solving the equation successfully.