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Chapter 15: Problem 83

Remove parentheses and simplify. $$ 6 x+5 y-7(x-y) $$

Short Answer

Expert verified
-x + 12y

Step by step solution

01

- Distribute the Negative Sign

First, remove the parentheses by distributing the negative sign (or the -7) inside the parentheses: - Apply -7 to both terms inside the parentheses:6x + 5y - 7(x - y) = 6x + 5y - 7x + 7y.
02

- Combine Like Terms

Now, combine the like terms. Group the x terms together and the y terms together: 6x - 7x + 5y + 7y.
03

- Simplify the Expression

Finally, simplify the expression by performing the arithmetic on the grouped terms: (6x - 7x) + (5y + 7y) = -x + 12y.

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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 key concept in algebraic simplification. This property allows you to distribute a number across terms inside parentheses. For example, in the expression \(a(b + c)\), you would use the distributive property to multiply \(a\) by both \(b\) and \(c\).
If you have a negative sign before the parentheses, as seen in the exercise \(6x + 5y - 7(x - y)\), you need to distribute -7 to both terms inside the parentheses.
This leads to: \(6x + 5y - 7 \times x + 7 \times y\).
So, you get \(6x + 5y - 7x + 7y\).
Combining Like Terms
Combining like terms is another crucial step in simplifying algebraic expressions. Like terms are terms that contain the same variable raised to the same power. For instance, \(3x\) and \(5x\) are like terms because they both contain \(x\) raised to the first power.
In the simplified version of the previous step \(6x + 5y - 7x + 7y\), you can identify the like terms easily: \(6x\) and \(-7x\) are like terms, and \(5y\) and \(7y\) are like terms.
Combining them, you get: \(6x - 7x + 5y + 7y\).
Negative Sign Distribution
Dealing with negative signs in algebra can be tricky. However, distributing negative signs correctly is essential for accurate algebraic simplification. When you have an expression like \(-a(b + c)\), you distribute the negative sign to both \(b\) and \(c\). This means you change the signs of both terms inside the parentheses.
In the exercise \(6x + 5y - 7(x - y)\), the negative sign in front of the \(7(x - y)\) indicates that you need to distribute \(-7\) to both \(x\) and \(-y\), leading to: \(6x + 5y - 7x + 7y\).
Remember, a negative sign multiplied by a negative term will become positive!
Algebraic Simplification
Algebraic simplification involves making an expression as simple as possible. After distributing and combining like terms, your goal is to have a reduced form of the original expression.
From our example, after distributing \(-7\) and combining like terms, we have the expression: \(6x - 7x + 5y + 7y\).
Perform the arithmetic by simplifying each group of like terms: \(6x - 7x = -x\) and \(5y + 7y = 12y\).
Thus, the simplified expression is \(-x + 12y\).
Always ensure you perform all operations accurately to achieve the most simplified form.

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