Problem 6 - Algebra for College Students

Question

Solve each equation. \(8-x=-2\)

Step-by-Step Solution

Answer

The solution is \(x = 10\).

Step 1: Identify the Equation

The given equation is \(8 - x = -2\). Our task is to solve for \(x\).

Step 2: Isolate the Variable

To solve for \(x\), we need to isolate it on one side of the equation. We do this by removing 8 from the left side. To do this, subtract 8 from both sides of the equation: \[ 8 - x - 8 = -2 - 8 \]Simplifying this, we get:\[ -x = -10 \]

Step 3: Solve for x

Now, multiply both sides by \(-1\) to solve for \(x\): \[ -1 imes (-x) = -1 imes (-10) \]Simplifying this, we get:\[ x = 10 \]

Key Concepts

Solving Equations

Equations are like a mathematical statement that shows the equality between two expressions. When solving algebraic equations, the main goal is to find the value of the unknown variable that makes the equation true. This process helps to understand the relationships between different quantities and how they interact with each other.

• Always start by identifying the equation and its components. Here, we have the equation: \(8 - x = -2\).
• Your task is to determine the value of \(x\) that satisfies this equation.

We solve equations using a set of steps that help systematically reach the solution. These steps usually involve mathematical operations such as addition, subtraction, multiplying, or dividing. Each step simplifies the equation until the unknown is isolated and can be calculated.

Isolating Variables

The strategy of isolating variables is central in solving equations. It involves rearranging the equation so that the variable you are solving for stands alone on one side of the equation. This helps in easily identifying the value of that variable.

• To isolate a variable, perform mathematical operations to eliminate other numbers or variables from the same side.
• In our example, start with: \(8 - x = -2\).
• We subtract 8 from both sides to isolate \(-x\) on one side: \((-x = -2 - 8)\).

By making sure you perform the same operation on both sides of the equation, you maintain the equality and make logical progress toward solving the equation.

Equation Simplification

Simplifying equations is a crucial step to make them easier to handle and solve. It involves combining like terms, reducing fractions, or performing mathematical operations that streamline the process.

• After isolating \(-x = -10\), the next step is to simplify to find \(x\).
• Multiply both sides by \(-1\) to change the sign and make \(x\) positive: \((-1 \times (-x) = -1 \times (-10))\).

Simplification not only makes equations easier but also reduces the possibility of errors in computation. The final result here is \(x = 10\), confirming that the steps led correctly to the solution.