What should be subtracted from 5y – 13x – 8a to obtain 11x – 16y + 7a?

Question

Accepted Answer

To determine what should be subtracted from the expression $5y - 13x - 8a$ to obtain $11x - 16y + 7a$, we need to find the difference between these two expressions. This involves setting up an equation where the expression to be subtracted, when subtracted from the original expression, results in the target expression.

Let's denote the expression to be subtracted as $E$. We have:

$$
(5y - 13x - 8a) - E = 11x - 16y + 7a
$$

Rearranging the equation to solve for $E$, we get:

$$
E = (5y - 13x - 8a) - (11x - 16y + 7a)
$$

Simplifying the right-hand side:

$$
E = 5y - 13x - 8a - 11x + 16y - 7a
$$

Combine like terms:

$$
E = (5y + 16y) + (-13x - 11x) + (-8a - 7a)
$$

$$
E = 21y - 24x - 15a
$$

Thus, the expression that should be subtracted is $21y - 24x - 15a$, which corresponds to Option 3.

**Explanation for why Option 3 is correct:**

- **Option 3: $21y - 24x - 15a$** is the correct expression to subtract because, when subtracted from the original expression $5y - 13x - 8a$, it results in the target expression $11x - 16y + 7a$. The calculations above show that this option correctly balances the equation.

**Explanation for why the other options are incorrect:**

- **Option 1: $6x + 21y + 15a$** is incorrect because substituting this into the equation would not yield the target expression. The coefficients do not match the required transformation from the original to the target expression.

- **Option 2: $21y - 5x - a$** is incorrect because the coefficients for $x$ and $a$ do not align with the necessary changes to transform the original expression into the target expression. The subtraction would not result in the correct target expression.

- **Option 4: $24x - 21y + a$** is incorrect because the signs and coefficients do not match the necessary transformation. Subtracting this option would not yield the target expression, as the coefficients for $x$, $y$, and $a$ are not correctly aligned.

In conclusion, Option 3 is the only choice that, when subtracted from the original expression, results in the desired target expression.