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Chapter 2: Problem 49

Solve the equations. More than one step is needed in each case.$$-2-3 x=-20$$

Short Answer

Expert verified
The solution is \(x = 6\).

Step by step solution

01

Isolate the variable term

To begin solving the equation \(-2 - 3x = -20\), we need to isolate the term that contains the variable \(x\). First, we will add 2 to both sides of the equation to move the constant term to the right side. This gives us:\[-3x = -20 + 2\].Simplifying the right side results in:\[-3x = -18\].
02

Solve for x

Next, we solve for \(x\) by dividing both sides of the equation by -3. This is to remove the coefficient of \(x\), which is -3:\[x = \frac{-18}{-3}\].Simplify the right side by performing the division:\[x = 6\].

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Key Concepts

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

Isolating the Variable
When tackling a linear equation, one of the first crucial steps is to isolate the variable you want to solve for. This means getting the variable, let's say \(x\), on one side of the equation all by itself. By doing this, you're essentially preparing the equation to be solved directly.

Take for instance, the equation \(-2 - 3x = -20\). The variable \(x\) is currently attached to the term \(-3x\) and is mixed up with other numbers. So, how do you isolate \(x\)?
  • Notice that \(-2\) is on the same side as \(-3x\). This is a constant that needs to be shifted to the other side of the equation.
  • To move \(-2\), add 2 to both sides of the equation. This balances the equation while leaving \(-3x\) alone. Now the equation reads: \(-3x = -18\).
Congratulations, now \(x\) is all by itself on the one side but still needs a little more work before we know its value. By isolating the variable, you are closer to the final solution.
Equation Simplification
Simplification is about making the equation easier to work with by performing specific mathematical operations. Once you've isolated the variable term, this is the next logical step. You'll adjust the numbers or terms on either side to make things straightforward.

In the equation \(-3x = -20 + 2\), you notice that there's an arithmetic operation that you can simplify immediately. This not only makes the numbers neater but also more manageable.
  • Perform the addition to combine numbers on the right side of the equation: \(-20 + 2\) results in \(-18\).
  • So, the simplified equation is \(-3x = -18\).
By conducting this simplification, the problem becomes clearer, showing you exactly what operations are necessary to finish solving for \(x\). In many cases, simplification helps prevent mistakes when you execute the final solution steps.
Coefficient Division
The final stage in solving a simple linear equation like this is to deal with the coefficient of the variable. In our example \(-3x = -18\), \(-3\) is the coefficient of \(x\). A coefficient is a number multiplying a variable, and to solve for \(x\), you need to eliminate this coefficient.

Since \(x\) is multiplied by \(-3\), the way to get \(x\) by itself is to perform the inverse operation, which is division.
  • Divide both sides of the equation by \(-3\) to solve for \(x\). This operation looks like: \(x = \frac{-18}{-3}\).
  • Carry out the division: \(-18 \div -3\) which gives \(x = 6\).
Now, \(x\) is finally isolated and solved. Division of the coefficient is often the last step in these equations, turning the variables into explicit values.

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Most popular questions from this chapter

Solve the equations by using the division property. $$-4 x=-24$$

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