Chapter 12: Problem 1
If
Short Answer
Expert verified
The value of x is 1.
Step by step solution
01
- Distribute and Simplify
First, distribute the constants through the parentheses. For the left side: 1) Expand: - Distribute 3: 3(x - 6) + 8 - (2 - 4x) = 3x - 18 + 8 - 2 + 4x. For the right side: 2) Distribute -4: 7 - 4(x + 2) = 7 - 4x - 8.
02
- Combine Like Terms
Combine like terms on both sides of the equation: For the left side: 3x - 18 + 8 - 2 + 4x = 7x - 12. For the right side: 7 - 4x - 8 = -4x - 1.
03
- Move All Terms Involving x to One Side
Add 4x to both sides of the equation to combine the x terms: 7x - 12 + 4x = -4x - 1 + 4x 11x - 12 = -1.
04
- Isolate the Variable
Solve for x by isolating the variable. First, add 12 to both sides: 11x - 12 + 12 = -1 + 12 . Thus: 11x = 11. Finally, divide both sides by 11: x = 1.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributive Property
When solving linear equations, one key concept is the distributive property. This property allows you to distribute a single term across terms inside parentheses. For instance, in the equation , we apply the distributive property to simplify:
Applying the distributive property helps to eliminate parentheses, turning the original equation into a line of terms: .
- First, distribute 3 through
: . - In the term
, distribute the negative sign, which changes signs of the terms inside the parentheses: . - On the right side, distribute -4 through
: .
Applying the distributive property helps to eliminate parentheses, turning the original equation into a line of terms:
Combining Like Terms
After using the distributive property, the next step in solving our linear equation is to combine like terms. Like terms are terms that contain the same variables raised to the same power.
Now, the simplified equation looks like this: . Combining like terms helps consolidate the equation, making it easier to solve.
- On the left side of our equation, combine the terms with
: . - Next, combine the constant terms:
. - The left side simplifies to
. - On the right side, combine the constant terms:
.
Now, the simplified equation looks like this:
Isolating Variable x
To solve for , we need to get all the terms on one side of the equation. Follow these steps:
Isolating the variable helps create a straightforward equation to solve for the variable.
- Add
to both sides of the equation to move the terms to one side: . - Simplify to get:
.
Isolating the
Algebraic Manipulation
The final step involves basic algebraic manipulation to isolate .
First, add 12 to both sides to move the constant term:
Combining all these steps, you can solve the equation easily by isolating the variable and applying basic algebraic operations.
First, add 12 to both sides to move the constant term:
.- This simplifies to:
.
-
- Which simplifies to
.
Combining all these steps, you can solve the equation easily by isolating the variable and applying basic algebraic operations.