Which binomial is equivalent to $3(x-1)-2(x-3)$?
We are given the algebraic expression \(3(x-1) - 2(x-3)\) and we need to simplify it to identify the correct equivalent binomial from the provided choices. To do this, we will use the distributive property, which states that \(a(b+c) = ab + ac\), and then combine the resulting like terms. Like terms are terms that have the same variable part, in this case, terms with \(x\) and constant terms.
First, apply the distributive property to the first part of the expression, \(3(x-1)\). Multiplying 3 by both \(x\) and \(-1\), we get:
\[3(x-1) = 3x - 3\]
Next, apply the distributive property to the second part, \(-2(x-3)\). It is important to remember to multiply by \(-2\), which will flip the sign of the constant term inside the parenthesis:
\[-2(x-3) = -2x + (-2 \times -3) = -2x + 6\]
Combining these two results, the expression becomes:
\[(3x - 3) + (-2x + 6)\]
Now, group the like terms together. The like terms are the terms containing \(x\) and the constant terms:
\[(3x - 2x) + (-3 + 6)\]
Simplifying the variable terms:
\[3x - 2x = x\]
Simplifying the constant terms:
\[-3 + 6 = 3\]
Adding these together gives:
\[x + 3\]
Comparing our simplified expression \(x+3\) with the given options:
The simplified expression matches option D. Therefore, the equivalent binomial is \(x+3\).
In daily life, the distributive property is used in scenarios like calculating the cost of multiple items. For example, if you are buying 3 packs of pens at \(\$(x-1)\) each and then returning 2 packs at \(\$(x-3)\) each, the total change in your budget is represented by the expression \(3(x-1) - 2(x-3)\). Simplifying this helps you easily determine that your net cost changes by \(x+3\) dollars, showing how algebraic simplification makes managing finances or inventory much faster.