8 | $4+5(3 x)$ |
9 | $-4(x-9)-3+7 x$ |
10 | $\begin{array}{l} 16 x+27+4 x-40 |
9(2 x+3)+4(x-10) | |
22 x+ \end{array}$ | |
11 | $6(x-7)-2(x+5)$ |
We are tasked with simplifying four sets of algebraic expressions. To simplify an expression, we use the distributive property, \(a(b+c) = ab + ac\), and then combine like terms. Like terms are terms that have the same variables raised to the same powers. We will treat each numbered item individually.
For expression 8: \(4+5(3x)\).
First, apply the distributive property to the term \(5(3x)\), which gives \(5 \times 3x = 15x\).
The expression becomes \(4 + 15x\). Since 4 and \(15x\) are not like terms, the expression cannot be simplified further. Thus, \(4 + 15x\).
For expression 9: \(-4(x-9)-3+7x\).
First, distribute \(-4\) into the parenthesis \((x-9)\): \(-4 \times x = -4x\) and \(-4 \times -9 = 36\).
The expression is now \(-4x + 36 - 3 + 7x\).
Group the like terms (the \(x\)-terms and the constants): \((-4x + 7x) + (36 - 3)\).
This simplifies to \(3x + 33\).
For item 10, we have three parts.
Part A: \(16x+27y+4x-40\). Combine the \(x\)-terms: \(16x + 4x = 20x\). The expression is \(20x + 27y - 40\).
Part B: \(9(2x+3)+4(x-10)\).
Distribute: \(9 \times 2x = 18x\), \(9 \times 3 = 27\), \(4 \times x = 4x\), and \(4 \times -10 = -40\).
The expression is \(18x + 27 + 4x - 40\).
Combine like terms: \((18x + 4x) + (27 - 40) = 22x - 13\).
Part C: The incomplete expression is \(22x + \dots\). Based on the simplification of Part B, it corresponds to \(22x - 13\).
For expression 11: \(6(x-7)-2(x+5)\).
Distribute: \(6 \times x = 6x\), \(6 \times -7 = -42\), \(-2 \times x = -2x\), and \(-2 \times 5 = -10\).
The expression is \(6x - 42 - 2x - 10\).
Combine like terms: \((6x - 2x) + (-42 - 10) = 4x - 52\).
Algebraic simplification is essential in fields like engineering and economics. For example, if you are calculating the total cost of a production line where 'x' represents units produced, you might have fixed costs and variable costs. Simplifying the expression allows you to quickly determine the base cost and the cost per unit, making it easier to predict expenses as production scales. Understanding how to group and combine terms helps streamline these business models effectively.