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Chapter 3: Problem 37

Solve the equation. $$-7 x+6(x+8)=-2$$

Short Answer

Expert verified
The solution is \(x = 50\).

Step by step solution

01

Distribute the 6

First, distribute the 6 through the expression in the parentheses. Rewrite the equation as follows: \(-7x + 6 \times (x + 8) = -2\) becomes \(-7x + 6x + 48 = -2\).
02

Combine Like Terms

Next, combine the like terms on the left side of the equation. The terms \(-7x\) and \(6x\) are like terms, so combine them:\(-7x + 6x = -1x\).This simplifies the equation to:\(-1x + 48 = -2\).
03

Isolate the Variable Term

Subtract 48 from both sides of the equation to isolate the variable term. \(-1x + 48 - 48 = -2 - 48\) simplifies to:\(-1x = -50\).
04

Solve for x

Divide both sides by -1 to solve for \(x\). \(\frac{-1x}{-1} = \frac{-50}{-1}\) gives:\(x = 50\).

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a fundamental concept used to simplify expressions in algebra. It helps in removing parentheses by distributing a multiplier over the terms inside the brackets. This can be described with the general formula: \[ a(b + c) = ab + ac \] In the given exercise, the number 6 needs to be distributed through the expression \((x + 8)\). Here's how it works step by step:
  • Multiply 6 by \(x\) resulting in \(6x\)
  • Multiply 6 by 8, which equals to 48
This transforms the equation from \(-7x + 6(x + 8) = -2\) into \(-7x + 6x + 48 = -2\). This is a crucial first step to simplify and prepare for further operations.
Combining Like Terms
Combining like terms is a process of merging terms that have the same variable raised to the same power. This simplification makes equations easier to solve. In our equation \(-7x + 6x + 48 = -2\), the like terms are \(-7x\) and \(6x\). They both have the variable \(x\) and can be combined by adding their coefficients:
  • \(-7x + 6x = -1x\)
Updating the equation gives us \(-1x + 48 = -2\). This step simplifies the equation drastically, leading us closer to the goal of solving for \(x\).
Solving Linear Equations
Solving linear equations involves finding the value of the unknown variable that makes the equation true. Once like terms are combined in \(-1x + 48 = -2\), the next task is to isolate \(x\). To achieve this, follow these steps:
  • Subtract 48 from both sides to remove the constant term: \(-1x + 48 - 48 = -2 - 48\)
  • This results in \(-1x = -50\)
  • Finally, divide both sides by \(-1\) to solve for \(x\): \(\frac{-1x}{-1} = \frac{-50}{-1}\)
  • The solution is \(x = 50\)
This process shows how systematic manipulation can solve equations, highlighting the importance of performing equivalent operations on both sides of the equation to maintain balance and solve for the unknown.

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