- Distribute the coefficients into the terms inside the parentheses, noting the signs: \(-4(5x+2) = -20x - 8\) and \(-6(x-3) = -6x + 18\).
- Combine like terms to reach the simplified form: \(-20x - 8 - 6x + 18 = -26x + 10\).
- Evaluate the original and simplified expressions at \(x=2\) to confirm equivalence, both resulting in \(-42\).
- Final simplified expression: \(\boxed{-26x+10}\). Value at \(x=2\): \(\boxed{-42}\).
Explanation
- Simplifying the Expression
We are given the algebraic expression \(-4(5x + 2) - 6(x - 3)\). To simplify this, we apply the distributive property, which states that \(a(b + c) = ab + ac\). We must carefully handle the signs when multiplying.
First, distribute the \(-4\) across \((5x + 2)\):
\[-4 \cdot 5x = -20x\]
\[-4 \cdot 2 = -8\]
So, \(-4(5x + 2) = -20x - 8\).
Next, distribute the \(-6\) across \((x - 3)\):
\[-6 \cdot x = -6x\]
\[-6 \cdot (-3) = 18\]
So, \(-6(x - 3) = -6x + 18\).
Combining these results, we get:
\[-20x - 8 - 6x + 18\]
Now, group the like terms (the \(x\) terms together and the constant terms together):
\[(-20x - 6x) + (-8 + 18) = -26x + 10\]
Thus, the simplified expression is \(-26x + 10\).
- Verification at \(x=2\)
To verify that the simplified expression \(-26x + 10\) is equivalent to the original expression \(-4(5x + 2) - 6(x - 3)\), we evaluate both at \(x = 2\).
For the original expression:
\[-4(5(2) + 2) - 6(2 - 3)\]
\[= -4(10 + 2) - 6(-1)\]
\[= -4(12) + 6\]
\[= -48 + 6\]
\[= -42\]
For the simplified expression:
\[-26(2) + 10\]
\[= -52 + 10\]
\[= -42\]
Both evaluations yield \(-42\), confirming that the expressions are equivalent.
- Conclusion
The simplified expression is \(-26x + 10\), and the value of both expressions at \(x=2\) is \(-42\). The simplified expression is \(\boxed{-26x+10}\) and the value at \(x=2\) is \(\boxed{-42}\).
Examples
Simplifying expressions is a fundamental skill in algebra that appears in various real-world scenarios. For example, if you are managing a budget where costs depend on variables (like the number of items purchased), you might have an expression like \(C = -4(5x+2) - 6(x-3)\) representing your net profit or loss. By simplifying this to \(C = -26x + 10\), you can quickly calculate how your budget changes as you adjust the quantity \(x\), allowing for faster decision-making without having to perform multiple complex calculations every time.