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Chapter 3: Problem 26

Solve each equation. $$-9 x+5=-18$$

Short Answer

Expert verified
The solution is \(x = \frac{23}{9}\).

Step by step solution

01

Eliminate Constant Term

To solve the equation \(-9x + 5 = -18\), we first need to isolate the term containing the variable, which is \(-9x\). We do this by eliminating the constant term on the left side of the equation. To do this, subtract 5 from both sides of the equation:\(-9x + 5 - 5 = -18 - 5\)This simplifies to:\(-9x = -23\).
02

Solve for x

Now that we have isolated the term with the variable \(-9x = -23\), we need to solve for \(x\). This requires getting \(x\) by itself on one side of the equation. To achieve this, divide both sides by \(-9\):\(\frac{-9x}{-9} = \frac{-23}{-9}\)This simplifies to:\(x = \frac{23}{9}\).

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Key Concepts

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

Isolating the Variable
When solving linear equations, the main goal is to isolate the variable, meaning we want the variable on one side of the equation by itself. This allows us to see the solution more clearly, as the variable represents the unknown we are trying to determine. In the equation, -9x + 5 = -18, the variable is x, and it is mixed with other terms.
  • Identify where your variable is placed, for example, here it's part of the term -9x.
  • The technique involves moving everything else off the side of the equation where your variable is, so x is all alone.
By following these steps, you make the equation clearer, as x will end up being explicitly expressed in terms of known numbers, allowing an easy solution.
Eliminating Constant Terms
Before isolating the variable, it's crucial to eliminate any constant terms that are influencing the equation. Constant terms are just plain numbers without any variables attached to them in the equation. In our equation, we have a constant term +5 that is on the same side as the variable term -9x. Since our goal is to get x alone, we must remove this constant by performing the inverse operation.
  • For example, to eliminate +5, you subtract 5 from both sides.
  • This method ensures whatever operation is done on one side is always mirrored on the other side to maintain equality throughout the process.
  • After subtraction, the modified equation becomes -9x = -23, effectively eliminating the constant term.
Eliminating constants brings the equation a step closer to extracting the variable cleanly.
Division in Equations
Once you have an equation where the variable term is isolated, like -9x = -23, your next step is to perform division to solve for the variable. Here, x is still not completely isolated because of the coefficient -9 that is multiplied with it.To make x independent, you will need to cancel out this coefficient by dividing both sides of the equation by the same number:
  • Dividing -9x by -9 results in x, since any number divided by itself is 1.
  • You must also divide the entire right side -23 by -9 to keep the equation balanced.
Performing this division gives us:\[x = \frac{23}{9}\]By finishing this step, you get the solution and can confidently state the value of x in its simplest form.

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