Which value of $y$ will make the equation true?
$-7y + 10 = -(5y - 6)$
The problem asks for the value of \(y\) that satisfies the equation:
\[-7y + 10 = -(5y - 6)\]
This is a linear equation in one variable. To find the solution, we need to simplify both sides and isolate \(y\). We start by distributing the negative sign on the right side of the equation. Distributing the negative sign means multiplying each term inside the parentheses by \(-1\). Thus, \(-(5y - 6) = -5y + 6\). Now the equation becomes:
\[-7y + 10 = -5y + 6\]
Now, we rearrange the equation to bring all terms containing \(y\) to one side and the constant terms to the other side. To remove \(-7y\) from the left, we add \(7y\) to both sides of the equation:
\[10 = 2y + 6\]
Next, to isolate the term containing \(y\), we subtract 6 from both sides of the equation:
\[10 - 6 = 2y\]
\[4 = 2y\]
Finally, to solve for \(y\), divide both sides by 2:
\[y = \frac{4}{2} = 2\]
We have found that \(y = 2\). We can verify this result by substituting \(y = 2\) back into the original equation:
LHS: \(-7(2) + 10 = -14 + 10 = -4\)
RHS: \(-(5(2) - 6) = -(10 - 6) = -(4) = -4\)
Since the left-hand side equals the right-hand side, our solution \(y = 2\) is correct. This matches the first option provided in the question.
This type of equation solving is fundamental in real-world scenarios, such as budgeting or comparing pricing models. For instance, if you are comparing two service plans, one might cost a base fee plus a rate per hour, and the other might have a different structure. By setting these expressions equal to each other, you can solve for the specific number of hours where both plans result in the same cost, allowing you to choose the more cost-effective option for your needs.