- Distribute the constants into the parentheses to get \(3x + 12 + 4x - 6\).
- Combine like terms to arrive at the simplified expression \(7x+6\).
- Substitute \(x=3\) into the original expression: \(3(3+4) + 2(2(3)-3) = 27\).
- Substitute \(x=3\) into the simplified expression: \(7(3)+6 = 27\).
- The simplified expression is \(\boxed{7x+6}\), and the value at \(x=3\) is \(\boxed{27}\).
Explanation
- Simplifying the Expression
To simplify the expression \(3(x+4) + 2(2x-3)\), we use the distributive property. First, distribute the \(3\) into the first set of parentheses:
\[3 \times x + 3 \times 4 = 3x + 12\]
Next, distribute the \(2\) into the second set of parentheses:
\[2 \times 2x - 2 \times 3 = 4x - 6\]
Adding these two results together, we get:
\[(3x + 12) + (4x - 6) = 3x + 4x + 12 - 6\]
Combining the like terms, the \(x\)-terms (\(3x\) and \(4x\)) and the constants (\(12\) and \(-6\)), we obtain the simplified expression:
\[7x + 6\]
- Verifying with \(x=3\)
To verify the simplification, we substitute \(x=3\) into both the original and the simplified expressions.
For the original expression:
\[3(3+4) + 2(2(3)-3) = 3(7) + 2(6-3) = 21 + 2(3) = 21 + 6 = 27\]
For the simplified expression \(7x+6\):
\[7(3) + 6 = 21 + 6 = 27\]
Since both expressions yield the same result, the simplification is correct. The simplified expression is \(7x+6\), and the numerical result for \(x=3\) is \(27\).
Examples
This type of algebraic simplification is essential in daily life for tasks like budgeting. For example, if you are buying 3 packs of apples (each containing \(x\) apples plus 4 extra) and 2 packs of oranges (each containing \(2x\) oranges minus 3), you can simplify the total count of fruits using this method. Simplifying expressions helps in managing resources efficiently and understanding how different variables affect total quantities.