91Ó°ÊÓ

To solve a linear equation, our first goal is to isolate the variable term on one side of the equation. This means we want the term with the variable (like 5x in our given problem) to be by itself on one side. Let's look at the equation again:
$$5x + 7 = 0$$
In this equation, the variable term is 5x. To isolate it, we need to get rid of the 7 that is on the same side as the variable term. We do this by performing the opposite operation of what is currently being done to the variable term. Since 7 is being added to 5x, we subtract 7 from both sides of the equation:
\ 5x + 7 - 7 = 0 - 7 \ This simplifies to: \ 5x = -7 \
By subtracting 7 from both sides, we have isolated the variable term 5x. This is an essential first step and simplifies the problem considerably. We are now ready to move to the next part: solving for x.

Now that we have isolated the variable term, the equation is simplified to:
\[5x = -7 \]
Our objective now is to solve for x, the variable. Since 5x means 5 times x, the opposite operation is division. We divide both sides of the equation by 5 to get x by itself. Here’s how we do it:
\[ \frac{5x}{5} = \frac{-7}{5} \]
Simplifying both sides, we get:
\[ x = -\frac{7}{5} \]
And just like that, we've solved for x! The solution to the equation is x = -\frac{7}{5}. This step is crucial because it reveals the value of x that satisfies the equation. All that remains is to verify this solution to ensure it is correct.

To make sure our solution for x is correct, we substitute it back into the original equation:
\[5x + 7 = 0 \]
Replace x with -\frac{7}{5} :
\[5 \left(-\frac{7}{5}\right) + 7 = 0 \]
Now, calculate the left side:
\[-7 + 7 = 0 \]
The left side simplifies to 0, which matches the right side of the original equation (0). This confirms that our solution x = -\frac{7}{5} is correct.
Checking the solution is an important step because it verifies the answer is accurate. If the two sides of the equation do not match, it means there was a mistake in the calculation. Always take the time to check your solutions to ensure correctness.