Please help: simple algebra confusion.

In summary, when dealing with a minus sign before a term, it should be thought of as part of the term rather than separating two terms. This can be illustrated by using substitution with parentheses to correctly remove the parentheses in an expression. Additionally, the statement can also be written as addition of signed numbers.
  • #1
Holocene
237
0
I’m currently on working on solving linear equations, and I’m having some trouble understanding whether a minus sign is actually separating two terms, or if it is actually part of the term.

Here’s something that snagged me, it’s the left member of the equation:

[tex]\displaystyle{7(2x - 3) - 4(x + 5)}[/tex]

Regarding the minus sign just before the “4”, should I think of that as “separating” the two terms, or should I think of it as actually being a part of the term, and hence the value is -4?

I keep wanting to write: [tex]\displaystyle{14x - 21 - 4x + 20}[/tex], but apparently this is false.

It should be [tex]\displaystyle{14x - 21 - 4x - 20}[/tex]

I’m just confused, because I clearly remember a rule stating that when subtracting a term, every sign has to change. For example:

[tex]\displaystyle{(7 - 3) - (2 - 1) = 7 - 3 - 2 + 1}[/tex]

Any help would be greatly appreciated!
 
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  • #2
[tex]x+5[/tex] can be thought of as one term, so it's the same as if you were multiplying 4 by any regular number or variable. Because it's one term you multiply the whole thing by 4, which gives you [tex]-4x-20[/tex].

That's how I like to think of it, hope that helped.
 
  • #3
Holocene said:
I’m currently on working on solving linear equations, and I’m having some trouble understanding whether a minus sign is actually separating two terms, or if it is actually part of the term.

Here’s something that snagged me, it’s the left member of the equation:

[tex]\displaystyle{7(2x - 3) - 4(x + 5)}[/tex]

Remember now the basic substitution rule:
Whenever substituting something by means of an identity, you will always get the correct result by setting a parenthesis around the new substitute.

Thus, since 4(x+5)=4x+20 and 7(2x-3)=14x-21, we may write:

7(2x-3)-4(x+5)=(14x-21)-(4x+20)=14x-21-4x-20, by removing the parentheses in the correct manner.

Since substitution with parantheses guarantees you the correct result (assuming you know your identities and how to remove parentheses), a good rule of thumb is always to substitute WITH parenthesis into a "complicated" expression.
 
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  • #4
Thanks everyone.
 
  • #5
You could think of it as an operation in subtraction of the qualtity 4(x+5), but the statement can also be written as 7(2x-3) + (-4)(x+5) and consider it addition of signed numbers.
 

1. What is algebra?

Algebra is a branch of mathematics that deals with solving equations and manipulating mathematical expressions using symbols and variables.

2. Why is algebra important?

Algebra is important because it is the foundation for many other branches of mathematics, such as geometry and calculus. It is also used in various real-life applications, such as finance, engineering, and science.

3. What are some common algebraic operations?

Some common algebraic operations include addition, subtraction, multiplication, division, and exponentiation. These operations can be performed on numbers, variables, or a combination of both.

4. How do I solve an algebraic equation?

To solve an algebraic equation, you need to isolate the variable on one side of the equation by using the inverse operation. This means performing the opposite operation to both sides of the equation until the variable is left alone on one side.

5. What are the most common mistakes when doing algebra?

Some common mistakes when doing algebra include not following the order of operations, forgetting to distribute or combine like terms, and making calculation errors. It is important to double-check your work and use proper notation to avoid these mistakes.

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