Question

Carlos solved the equation below for x using inverse operations. His solution is x = 6.5. What mistake did Carlos make and how could he have checked his solution to know for certain that his answer is correct?
5x + 7.5 = 25
5x = 32.5
x = 6.5

Answer 1

How could Carlos have checked his solution? Show each step by plugging his solution for x back into the equation.

Answer 2

Let's go through the steps of the equation that Carlos solved to identify the mistake he made.

The original equation is:

\[ 5x + 7.5 = 25 \]

To isolate \( x \), you would first subtract \( 7.5 \) from both sides:

\[ 5x = 25 - 7.5 \]

Calculating the right side:

\[ 25 - 7.5 = 17.5 \]

So, the equation should become:

\[ 5x = 17.5 \]

Next, you would divide both sides by \( 5 \):

\[ x = \frac{17.5}{5} \]

Now calculating \( \frac{17.5}{5} \):

\[ x = 3.5 \]

So, Carlos made a mistake in his subtraction. Instead of correctly calculating \( 25 - 7.5 \), he incorrectly got \( 32.5 \).

To check his solution, he could substitute his value of \( x \) back into the original equation:

1. Substitute \( x = 6.5 \) into the left side of the original equation:

\[ 5(6.5) + 7.5 = 32.5 + 7.5 = 40 \]

2. Then check if this equals \( 25 \):

Since \( 40 \neq 25 \), Carlos would see that his answer was incorrect.

By substituting his solution back into the original equation, he would have recognized that \( x = 6.5 \) was not a valid solution, leading him to re-evaluate his calculations.

Answer 3

To check his solution, Carlos can substitute his value of \( x = 6.5 \) back into the original equation step by step. Here’s how he would do it:

### Original equation:
\[
5x + 7.5 = 25
\]

### Step 1: Substitute \( x = 6.5 \) into the equation:
\[
5(6.5) + 7.5 = 25
\]

### Step 2: Calculate \( 5(6.5) \):
\[
5 \times 6.5 = 32.5
\]

### Step 3: Add \( 7.5 \) to \( 32.5 \):
\[
32.5 + 7.5 = 40
\]

### Step 4: Compare the result to the right side of the equation:
The left side of the equation is now \( 40 \), while the right side is \( 25 \):

\[
40 \neq 25
\]

### Conclusion:
Carlos would have found that the left side does not equal the right side, demonstrating that his solution \( x = 6.5 \) is incorrect. This process of substituting the value back into the original equation is an effective way to verify whether a solution is valid.