Problem 62 - Prealgebra

Question

Solve each equation. Check your solution. $$10 x=300$$

Step-by-Step Solution

Answer

The solution is \(x = 30\).

Step 1: Isolate the variable

To solve the equation \(10x = 300\), we need to isolate \(x\). Since \(x\) is being multiplied by \(10\), we'll use the inverse operation, which is division. Divide both sides of the equation by \(10\): \[ x = \frac{300}{10} \]. This simplifies to \(x = 30\).

Step 2: Check the solution

Substitute \(x = 30\) back into the original equation to verify your solution. Replace \(x\) with \(30\) in the equation \(10x = 300\): \[ 10 \times 30 = 300\]. Since \(300 = 300\), our solution is verified as correct.

Key Concepts

Checking Solutions

Once we find a solution to an equation, the next step is to confirm whether it is correct. This process is known as checking the solution. To do this, we substitute the solution back into the original equation to see if the equation still holds true. For example, in the problem where we concluded that \( x = 30 \), we substitute \(x = 30\) back into the equation \( 10x = 300 \).

• Multiply: \(10 \times 30\) to get \(300\).
• If the left side equals the right side (i.e., \(300 = 300\)), then the solution is correct.

Checking solutions is crucial because it ensures that no mistakes were made during the process, and the equation is solved correctly.

Isolation of Variables

The goal when solving equations is to find the value of the unknown variable. To do this effectively, we often use a strategy called "isolation of variables." This means rearranging the equation so that the variable we are trying to solve for is by itself on one side of the equation. In the given exercise, the original equation is \(10x = 300\). To isolate \(x\), we need to remove the number that is multiplied by it. In this case, that number is 10. We achieve isolation by performing the same operation on both sides of the equation, which in this case is division:

• Divide both sides by 10 to isolate \(x\): \( x = \frac{300}{10} \).
• This simplifies to \(x = 30\), successfully isolating the variable.

Inverse Operations

Inverse operations are mathematical operations that undo each other. When solving for a variable, we often employ inverse operations to cancel out operations performed on the variable. In our exercise, the variable \(x\) is multiplied by 10. The inverse of multiplication is division, so we use it to solve the equation:

• To undo the multiplication in \(10x = 300\), we divide both sides by 10.
• This results in \(x = \frac{300}{10} \), which simplifies to \(x = 30\).

Utilizing inverse operations is essential in simplifying equations and finding the correct solutions. They are fundamental tools in the problem-solving process in algebra.