Problem 9 - Prealgebra

Question

Solve each equation. Check your solution. $$8 x=72$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 9\).

Step 1: Understand the Equation

The provided equation is \(8x = 72\). This is a simple linear equation where the goal is to find the value of \(x\) that makes the equation true.

Step 2: Isolate the Variable

To solve for \(x\), divide both sides of the equation by 8, which is the coefficient of \(x\): \[\frac{8x}{8} = \frac{72}{8}\]This simplifies to:\[x = 9\]

Step 3: Verify the Solution

To check if \(x = 9\) is correct, substitute \(x\) back into the original equation:\[8(9) = 72\]Calculate \(8 \times 9\) which equals 72. Since both sides of the equation are equal, \(x = 9\) is a correct solution.

Key Concepts

Solving Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. Solving these equations often requires you to find the value of the unknown variable that makes the equation true. In the equation \(8x = 72\), our goal is to isolate \(x\) so that it stands alone on one side of the equation.

• Identify the coefficient of the variable, which in this case is 8.
• To isolate \(x\), perform the opposite operation to what's being done to \(x\). Here, multiply is reversed by division.
• Divide both sides by the same number to maintain equality. This gives us \(\frac{8x}{8} = \frac{72}{8}\), simplifying to \(x = 9\).

Always ensure to perform the same operation on both sides. This is key to preserving the balance of the equation, leading you to the correct solution.

Checking Solutions

After finding a possible solution, it is crucial to check that the solution satisfies the original equation. This verifies the answer's correctness.

• Substitute the found value of the variable back into the original equation. For this exercise, replace \(x\) with 9 in \(8x = 72\).
• Calculate \(8 \times 9\), resulting in 72.
• Confirm both sides of the equation are equivalent.

If both sides match, then the solution is verified as correct. If not, errors may have occurred during calculations or the solving process.

Prealgebra Concepts

Prealgebra forms the foundation of understanding algebra by introducing basic math concepts. It includes working with numbers, understanding mathematical operations, and grasping equation-solving techniques.

• Comprehend how to manipulate equations using addition, subtraction, multiplication, and division.
• Learn the importance of maintaining equality by performing the same operations on both sides of an equation.
• Begin to recognize how different values interact, and how variables represent unknown numbers you need to discover.

Prealgebra's role is to enhance your fluency in math, preparing you for more complex algebraic challenges. The skills you acquire here are indispensable as they simplify future math courses by ensuring your foundational knowledge is strong.