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Chapter 2: Problem 60

Solve the given equation for \(\mathrm{x}\). \( .-19 x-17=-36\)

Short Answer

Expert verified
The solution is x = 1.

Step by step solution

01

Simplify the equation

Start by isolating the term with the variable on one side of the equation. To do this, add 17 to both sides of the equation:\[-19x - 17 + 17 = -36 + 17\]This simplifies to:\[-19x = -19\].
02

Solve for x

Now, solve for \(x\) by dividing both sides of the equation by -19:\[x = \frac{-19}{-19}\]This simplifies to:\[x = 1\].

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Key Concepts

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

Isolating Variables
Isolating variables is a key technique in solving linear equations. The goal is to manipulate the equation so that the variable of interest is alone on one side of the equation. This makes it easier to find its value. Here's how to do this effectively:
  • Identify the variable that needs to be isolated, in this case, it's \(x\).
  • Perform operations on both sides of the equation to remove any constants or coefficients from around the variable.
  • In the original equation \(-19x - 17 = -36\), we want to get \(-19x\) by itself. Adding 17 to both sides of the equation achieves this, modifying it to \(-19x = -19\).
This leaves the variable term clear of other numbers, simplifying the equation for easier solving. By keeping operations balanced on both sides, we ensure the equation remains equal.
Simplification
Simplification is the process of making an equation easier to work with. We achieve this by performing operations that reduce complexity. In our example, simplification includes two main steps:
  • Removing Additional Terms: By adding or subtracting terms, similar to how we added 17 to both sides to eliminate the \(-17\), we simplify the equation structure.
  • Reducing Fractions or Coefficients: This involves dividing or multiplying terms to simplify numbers or coefficients.
Once simplified to \(-19x = -19\), it's easier to see what operations are needed to solve for \(x\). Simplification transforms the equation into a form where basic operations can be directly applied, helping to find solutions more efficiently.
Division by Negative Numbers
Dividing by negative numbers can sometimes be confusing, but it's an important step in equations like the one we're solving. Here's how to handle it:
  • When you divide both sides of the equation by a negative number, remember that the signs will change.
  • In the case of \(-19x = -19\), dividing both sides by \(-19\) cancels out the negative coefficient attached to \(x\), giving us \(x = 1\).
This step is crucial to solving the equation correctly. It's essential to do this division carefully, ensuring both side operations are equivalent. The negative signs cancel each other out, which is why \(\frac{-19}{-19} = 1\). This demonstrates that division by a negative doesn't alter the problem's integrity but is pivotal in reaching the solution.

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Most popular questions from this chapter

Five more than eight times an unknown number is \(-35\). Find the unknown number.

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