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Chapter 3: Problem 46

Solve the equation. $$-8 x+7(x-2)=-14$$

Short Answer

Expert verified
The solution is \(x = 0\).

Step by step solution

01

Distribute the 7

Expand the expression on the left-hand side by distributing the 7 into the terms inside the parenthesis.The equation becomes:\(-8x + 7x - 14 = -14\).
02

Combine Like Terms

Combine the like terms on the left-hand side. Here, combine the coefficients of \(x\):\(-8x + 7x = -1x\).The equation simplifies to:\(-1x - 14 = -14\).
03

Isolate the Variable

To isolate the variable \(x\), add 14 to both sides of the equation:\(-1x - 14 + 14 = -14 + 14\).This simplifies to:\(-1x = 0\).
04

Solve for x

Divide both sides of the equation by -1 to solve for \(x\):\(x = \frac{0}{-1}\).This simplifies to:\(x = 0\).

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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 useful tool in algebra that helps us simplify expressions and solve equations. It involves multiplying a single term across terms inside a parenthesis. In the equation given, \(-8x + 7(x-2) = -14\), the distributive property allows us to take the 7 and multiply it by each term within the parenthesis. So, we multiply 7 by \(x\) and 7 by -2. This changes the equation to:
  • \(7 imes x = 7x\)
  • \(7 imes -2 = -14\)
Using the distributive property, the expression within the equation becomes \(-8x + 7x - 14\). This step gives us a simpler equation to work with, preparing it for the next step of combining like terms.
Combining Like Terms
The process of combining like terms involves merging terms that have the same variable component to simplify the expression further. In our equation, after applying the distributive property, we have:\(-8x + 7x - 14\).The terms \(-8x\) and \(7x\) are like terms because they both involve the variable \(x\). By adding these coefficients together (while keeping the variable), we perform the following calculation:
  • \(-8 + 7 = -1\)
This simplification step reduces the equation to \(-1x - 14 = -14\). It鈥檚 crucial during problem solving because it simplifies the equation, making it cleaner and leading us to the next key step of isolating the variable.
Isolating Variables
Isolating the variable is an essential step when solving equations. It involves getting the variable by itself on one side of the equation to directly solve for its value. In our current equation, \(-1x - 14 = -14\), we can isolate the variable \(x\) by eliminating the constant on the same side as \(x\). To remove \(-14\), we add 14 to both sides of the equation:
  • \(-1x - 14 + 14 = -14 + 14\)
This cancellation of \(-14\) results in \(-1x = 0\).To solve for \(x\), divide both sides of the equation by \(-1\) to obtain:
  • \(x = \frac{0}{-1}\)
  • \(x = 0\)
By isolating \(x\), we find the solution to the equation. This step is crucial because it transforms our simplified equation into a straightforward result, thus completing the process.

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