- Distribute the negative sign in the expression \((15 x^2+11 y^2+8 x)-(7 x^2+5 y^2+2 x)\).
- Combine like terms: \((15 - 7)x^2 = 8x^2\), \((11 - 5)y^2 = 6y^2\), and \((8 - 2)x = 6x\).
- Compare the result \(8 x^2+6 y^2+6 x\) to the original right-hand side \(A x^2+6 y^2+6 x\).
- Conclude that \(A = 8\), so the final answer is \(\boxed{8}\).
Explanation
- Analysis of the Equation
We are given the algebraic equation \((15 x^2 + 11 y^2 + 8 x) - (7 x^2 + 5 y^2 + 2 x) = A x^2 + 6 y^2 + 6 x\). Our objective is to determine the value of the coefficient \(A\) by simplifying the left-hand side of the equation and matching the coefficients of corresponding terms.
- Simplifying the Left-Hand Side
To simplify the expression on the left-hand side, we first distribute the negative sign into the second set of parentheses:
\[(15 x^2 + 11 y^2 + 8 x) - 7 x^2 - 5 y^2 - 2 x\]
Next, we group the like terms together:
\[(15 x^2 - 7 x^2) + (11 y^2 - 5 y^2) + (8 x - 2 x)\]
Performing the arithmetic for each group, we have:
\[(15 - 7) x^2 + (11 - 5) y^2 + (8 - 2) x = 8 x^2 + 6 y^2 + 6 x\]
- Identifying the Missing Coefficient
Now that we have the simplified form of the left-hand side, we compare it with the right-hand side of the original equation:
\[8 x^2 + 6 y^2 + 6 x = A x^2 + 6 y^2 + 6 x\]
By comparing the coefficients of the \(x^2\) terms on both sides, we can clearly see that \(A = 8\).
- Final Conclusion
Based on our comparison, the missing coefficient is 8, which corresponds to the second option provided. Thus, the value of \(A\) is \(\boxed{8}\).
Examples
In inventory management, you might need to calculate the change in stock levels. For example, if you start with 15 units of item X and 11 units of item Y, and you remove 7 units of X and 5 units of Y, this algebraic subtraction helps you track the final quantity remaining in your warehouse. Comparing coefficients in an expression is similar to balancing your accounts, where you ensure the items you started with, adjusted by subtractions, equal the total ending stock.