Problem 20 - Prealgebra

Question

Solve each equation. Check your solution. $$45=5 x$$

Step-by-Step Solution

Answer

The solution is \(x = 9\).

Step 1: Identify the Equation

The equation given to us is \(45 = 5x\). Our task is to solve for \(x\), which is the unknown value in the equation.

Step 2: Isolate the Variable

To solve for \(x\), we need to isolate \(x\) on one side of the equation. We can do this by dividing both sides of the equation by 5 to get \(x\) by itself. This gives us \(x = \frac{45}{5}\).

Step 3: Simplify the Expression

Now, we simplify the right-hand side of the equation. The expression \(\frac{45}{5}\) simplifies to 9. Thus, we have \(x = 9\).

Step 4: Check the Solution

To verify our solution, we substitute \(x = 9\) back into the original equation. We check if \(5 \times 9 = 45\). Calculating this gives us 45, which confirms that the solution is correct.

Key Concepts

Solving Equations

An equation is like a balance scale, with both sides needing to weigh the same. Here, we have the equation \(45 = 5x\). To solve it means finding the value of \(x\) that keeps both sides equal. To do this, you should:

• Understand what the equation is asking - it's the value of \(x\) that balances the equation.
• Identify the terms and constants on both sides. In this case, \(45\) is the constant and \(5x\) represents the five times \(x\).
• Use mathematical operations to "solve" the equation, keeping the balance in place.

Starting with the given equation provides a clear path for the subsequent steps needed to find the value of the unknown variable, \(x\). Remember, the aim is to manipulate the equation while maintaining equality.

Isolation of Variables

The process of isolation involves getting the variable \(x\) on one side of the equation by itself. With our example of \(45 = 5x\), we want to isolate \(x\). This is done through inverse operations.First, identify the operation affecting \(x\). Here, \(x\) is multiplied by \(5\). To isolate \(x\), you need to divide both sides of the equation by \(5\):

• The operation used is division, because it is the inverse of multiplication.
• This gives: \(x = \frac{45}{5}\).

This step is crucial, as it logically reduces the known from the equation, allowing \(x = 9\). Understanding this step helps build algebraic thinking, focusing on using inverse operations to clearly see how each step maintains the balance of the original equation.

Checking Solutions

Checking the solution is the final step to ensure the correctness of your answer. Once you believe you have the value of \(x\), substitute it back into the original equation to verify.In this exercise, we found \(x = 9\). To check:

• Substitute \(9\) back into the equation: \(5 \times 9\).
• This leads to the expression \(5 \times 9 = 45\).
• Since \(45 = 45\) holds true, the solution is confirmed as correct.

This step reassures that no mistakes were made during solving the equation. Verification is essential in mathematics as small errors can occur, and checking the solution thoroughly ensures correctness before concluding. It’s an important habit to get into when solving equations.