- Simplify the equation by combining like terms on the left side: \(2y + 7 = 2y + 70\).
- Subtract \(2y\) from both sides to isolate the constants: \(7 = 70\).
- Since \(7 = 70\) is a false statement, the equation has no solution.
- The correct option is (A).
- The final answer is \(\boxed{\text{No solutions}}\).
Explanation
- Analysis of the Equation
The equation given is \(-15y + 7 + 17y = 2y + 70\). We are asked to determine how many solutions it has. To solve this, we will simplify the expression on the left-hand side and then attempt to isolate the variable \(y\). An equation is considered to have no solutions if we reach a contradiction (a false statement), exactly one solution if we find a specific value for \(y\), and infinitely many solutions if the equation simplifies to a tautology (a statement that is always true).
- Simplifying the Expression
First, let's simplify the left-hand side (LHS) by combining like terms. The LHS is \(-15y + 7 + 17y\). Grouping the \(y\)-terms together, we have:
\[(-15y + 17y) + 7 = 2y + 7.\]
Now, we set the simplified LHS equal to the right-hand side (RHS) of the original equation:
\[2y + 7 = 2y + 70.\]
- Isolating the Variable and Conclusion
Next, we aim to isolate the variable \(y\) by gathering all terms involving \(y\) on one side of the equation. We can do this by subtracting \(2y\) from both sides of the equation:
\[(2y + 7) - 2y = (2y + 70) - 2y.\]
Simplifying both sides, we get:
\[7 = 70.\]
This is a statement that is clearly false for any value of \(y\). Since the assumption that a value of \(y\) exists leads to a mathematical contradiction, we conclude that there is no value of \(y\) that satisfies the equation.
Examples
Think of this problem like trying to find a balance point on a seesaw where the weights on both sides must be equal. If the equation simplifies to something impossible like \(7 = 70\), it means the weights can never be balanced, no matter where you position them. This is common in real-world scenarios, such as trying to reconcile two budgets that have inherently different, non-reconcilable constant costs, showing that no single solution can exist to make both sides equal.