Problem 60
Question
Simplify each algebraic expression. $$7(3 y+5)+(-25 y)$$
Step-by-Step Solution
Verified Answer
-4y+35
1Step 1: Distribute the 7
Distribute the 7 across the terms inside the parenthesis: \(7 \times 3y = 21y\) and \(7 \times 5 = 35\). So, our equation becomes \(21y+35-25y\).
2Step 2: Combine Like Terms
The next step is combining like terms. The terms are \(21y\) and \(-25y\). Combining these gives \(-4y\). Now you are left with \(-4y+35\).
3Step 3: Final Simplification
Once you have combined all like terms, the expression is fully simplified. In this case, there are no other like terms to combine so our simplified equation is \(-4y+35\).
Key Concepts
Distributive PropertyCombining Like TermsAlgebraic Simplification
Distributive Property
Understanding the distributive property is crucial in algebra. It's a tool that allows you to multiply a single term by each term inside a parenthesis. Let's imagine you're at a fruit stand buying bags of apples and oranges. One bag has 3 apples and 5 oranges. Instead of buying the fruits separately, you buy 7 bags. You're essentially multiplying the contents of one bag by 7.
In mathematics, this is akin to expanding expressions such as the one in our exercise: \(7(3y + 5)\). Here, we distribute the 7 to each term inside the parentheses: \(7 \times 3y\) gives us \(21y\), and \(7 \times 5\) results in 35. So after distribution, we have the expression \(21y + 35\). This process simplifies the evaluation of algebraic expressions and prepares the stage for further simplification.
In mathematics, this is akin to expanding expressions such as the one in our exercise: \(7(3y + 5)\). Here, we distribute the 7 to each term inside the parentheses: \(7 \times 3y\) gives us \(21y\), and \(7 \times 5\) results in 35. So after distribution, we have the expression \(21y + 35\). This process simplifies the evaluation of algebraic expressions and prepares the stage for further simplification.
Combining Like Terms
Once the distributive property has been applied, it's time to 'tidy up' the expression by combining like terms. Think of this as organizing a bookshelf by grouping together books of the same genre. In algebra, like terms are those that have the same variable raised to the same power. In the expression \(21y + 35 - 25y\), look for terms that have the same variable part.
Here, \(21y\) and \(-25y\) are like terms because they both have the variable \(y\). To combine them, simply add or subtract their coefficients: \(21 - 25\), which equals \(-4\). The \(y\) follows along unchanged, resulting in \(-4y\). So after combining like terms, our expression simplifies to \(-4y + 35\). This step helps to create a more concise and manageable expression that is easier to work with.
Here, \(21y\) and \(-25y\) are like terms because they both have the variable \(y\). To combine them, simply add or subtract their coefficients: \(21 - 25\), which equals \(-4\). The \(y\) follows along unchanged, resulting in \(-4y\). So after combining like terms, our expression simplifies to \(-4y + 35\). This step helps to create a more concise and manageable expression that is easier to work with.
Algebraic Simplification
The ultimate goal in simplifying algebraic expressions is to make them as straightforward as possible. To continue the bookshelf analogy, algebraic simplification is the final step where you place your sorted books neatly, ensuring the shelf is organized. After distributing and combining like terms, you should look for any final adjustments. In the given exercise, after combining like terms, we're left with \(-4y + 35\).
There are no more like terms to merge, meaning our expression is at its simplest form. Algebraic simplification can involve several rounds of distributing and combining like terms, especially with more complex equations. However, once you've done all that's possible, you're left with an elegantly simplified expression that conveys the same information as the original, but in a much more accessible way.
There are no more like terms to merge, meaning our expression is at its simplest form. Algebraic simplification can involve several rounds of distributing and combining like terms, especially with more complex equations. However, once you've done all that's possible, you're left with an elegantly simplified expression that conveys the same information as the original, but in a much more accessible way.
Other exercises in this chapter
Problem 59
Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{3}{4} \div \frac{1}{4}$$
View solution Problem 60
In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$130 \div(-10)$$
View solution Problem 60
Simplify each algebraic expression. $$2(5 x+4)-3$$
View solution Problem 60
Insert either \(\) in the shaded area between each pair of numbers to make a true statement. $$0.6\quad\square\quad0 . \overline{6}$$
View solution