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Chapter 1: Problem 29

Solve each equation, if possible. $$ 8 x-(3 x+2)=3 x-10 $$

Short Answer

Expert verified
The solution to the equation is \( x = -4 \).

Step by step solution

01

- Distribute and simplify both sides

First expand and simplify the equation. The given equation is: \[ 8x - (3x + 2) = 3x - 10 \]Distribute the negative sign on the left-hand side:\[ 8x - 3x - 2 = 3x - 10 \]
02

- Combine like terms

Next, combine the like terms on the left-hand side:\[ (8x - 3x) - 2 = 3x - 10 \]This simplifies to:\[ 5x - 2 = 3x - 10 \]
03

- Isolate the variable term

Move all terms containing \( x \) to one side and the constant terms to the other side. Subtract \( 3x \) from both sides:\[ 5x - 3x - 2 = 3x - 3x - 10 \]Simplify the equation:\[ 2x - 2 = -10 \]
04

- Solve for the variable

Add 2 to both sides to isolate the term containing \( x \):\[ 2x - 2 + 2 = -10 + 2 \]This simplifies to:\[ 2x = -8 \]Finally, divide both sides by 2 to solve for \( x \):\[ x = \frac{-8}{2} = -4 \]

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Key Concepts

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

distribution in algebra
Distribution in algebra involves multiplying each term inside a parenthesis by the factor outside. This process helps in simplifying algebraic expressions and equations. For example, in the equation \[ 8x - (3x + 2) = 3x - 10 \], we distribute the negative sign across the terms inside the parentheses:
  • First term: \[-(3x) = -3x\]
  • Second term: \[-2\]
After distributing, the equation becomes \[8x - 3x - 2 = 3x - 10\].Now, the equation is simpler, ready for the next steps of solving.
combining like terms
Combining like terms is an essential step in solving equations. Terms are 'like' if they have the same variable raised to the same power. In the equation \[8x - 3x - 2 = 3x - 10\], we combine the like terms on the left side:
  • Like terms: \[8x\] and \[-3x\]
  • Combined: \[8x - 3x = 5x\]
This simplifies the expression to \[5x - 2 = 3x - 10\]. Combining like terms helps in making the equation more manageable by reducing the number of terms.
isolating variables
Isolating the variable means getting the variable term alone on one side of the equation. In \[5x - 2 = 3x - 10\], we start by moving all \[x\] terms to one side and constants to the other. Subtract \[3x\] from both sides:
  • \[5x - 3x = 2x\]
Now we have \[2x - 2 = -10\]. Next, we move the constant term by adding \[2\] to both sides:
  • \[2x - 2 + 2 = -10 + 2\]
It simplifies to \[2x = -8\], making it easier to solve for \[x\].
simplifying equations
Simplifying equations involves performing operations to make an equation easier to solve. After isolating the variable term in \[2x = -8\], we need to get \[x\] alone by dividing both sides by \[2\]:
  • \[\frac{2x}{2} = \frac{-8}{2}\]
  • Final simplified term: \[x = -4\]
With \[x\] isolated and the equation simplified, we find the solution \[x = -4\]. Always check your final answer by plugging it back into the original equation to ensure both sides are equal.

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