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Chapter 1: Problem 72

Simplify each expression. $$ -4(5 y-7)+3(2 y-5) $$

Short Answer

Expert verified
-14y + 13

Step by step solution

01

- Distribute the Constants

Apply the distributive property to each term within the parentheses.\[ -4(5y - 7) + 3(2y - 5) \]This expands to:\[ -4 \times 5y + (-4 \times -7) + 3 \times 2y + 3 \times -5 \]Which simplifies to:\[ -20y + 28 + 6y - 15 \]
02

- Combine Like Terms

Combine the terms with variables and the constant terms separately. From the expanded expression:\[ -20y + 28 + 6y - 15 \]Combine the like terms \(-20y\) and \(6y\):\[ -20y + 6y = -14y \]Combine the constants 28 and -15:\[ 28 - 15 = 13 \]So, the expression simplifies to:\[ -14y + 13 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
The distributive property is a fundamental concept in algebra. It allows us to simplify expressions by multiplying each term inside a parenthesis by a factor outside the parentheses. This is how we distribute multiplication over addition or subtraction. Look at our expression: To apply the distributive property, we multiply each term inside the parentheses by the constant outside: This expands to: Within this step, be careful with the signs to avoid mistakes. When we distribute -4 and 3, we get:
combining like terms
Combining like terms is the process of simplifying an expression by summing or subtracting terms that have identical variable parts. For instance, in the simplified step above: We see terms -20y and 6y with the variable 'y'. Combine these two: Additionally, combine the constants 28 and -15. This simplifies the expression by reducing it to fewer terms, making it easier to understand and solve.
constants and variables
In algebra, it's essential to differentiate between constants and variables. A variable is a symbol representing an unknown value, usually a letter like 鈥榵鈥� or 鈥榶鈥�, while a constant is a fixed value like 2, 5, or -3. In our final simplified expression, we have -14y + 13 Here ( -14y ) contains the variable 'y', determining how the term changes with 'y's value. On the other hand, 13 is a constant, meaning it remains fixed irrespective of 'y'. By separating and combining them during simplification, we achieve a clearer and more concise expression.

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Most popular questions from this chapter

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) $$ \frac{x y+8 a}{x-y} $$

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 6 if it is divisible by both 2 and \(3,\) and not otherwise. Show that (a) \(1,524,822\) is divisible by 6 and \((b) 2,873,590\) is not divisible by 6

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To find the average (mean) of numbers, we add the numbers and then divide the sum by the number of terms added. For example, to find the average of \(14,8,3,9,\) and \(1,\) we add them and then divide by 5. $$ \frac{14+8+3+9+1}{5}=\frac{35}{5}=7 \leftarrow \text { Average } $$ Find the average of each group of numbers. \(23,18,13,-4,\) and \(-8\)

Simplify each expression. $$ -6-4(y-7) $$
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