Simplify this expression using distribution and combining like terms:
$4(2x - 3) - 3x$
To simplify the expression \(4(2x - 3) - 3x\), we first apply the distributive property, which states that \(a(b - c) = ab - ac\). Distributing the 4 gives us:
\[4(2x - 3) = (4 \times 2x) - (4 \times 3) = 8x - 12.\]
Now, we substitute this back into the original expression:
\[8x - 12 - 3x.\]
Next, we group the like terms (the terms containing \(x\)):
\[(8x - 3x) - 12.\]
Performing the subtraction for the coefficients of \(x\), we get \(8 - 3 = 5\), so the expression simplifies to:
\[5x - 12.\]
To simplify the second expression \(4(x - 3)\), we again apply the distributive property. Distributing the 4 across the terms inside the parentheses:
\[4(x - 3) = (4 \times x) - (4 \times 3).\]
This yields:
\[4x - 12.\]
Since there are no more like terms to combine, this is the final simplified form for the second expression.
By applying the distributive property and combining like terms, we have successfully simplified the two given expressions. The first expression \(4(2x - 3) - 3x\) simplifies to \(5x - 12\), and the second expression \(4(x - 3)\) simplifies to \(4x - 12\). Therefore, the simplified forms are \(5x - 12\) and \(4x - 12\), respectively.
Distributing and combining like terms is a foundational skill in algebra, useful for organizing data in real-world scenarios. For example, if a store sells two types of items where the first item costs \(\$2\) each and the second costs \(\$3\) each, and you buy a set of bundles containing these, you can calculate the total cost by creating an algebraic model. Simplifying this model helps determine the total expenditure efficiently, similar to how we simplified the expressions to understand their underlying structure.