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Chapter 2: Problem 45

Solve each equation. $$\frac{2}{3} x+5=-\frac{1}{3} x+17$$

Short Answer

Expert verified
x = 12

Step by step solution

01

Combine Like Terms

First, move all the terms involving the variable to one side of the equation. Add \(\frac{1}{3}x\) to both sides:\[ \frac{2}{3}x + \frac{1}{3}x + 5 = 17 \]
02

Simplify

Combine the like terms on the left side. \(\frac{2}{3}x + \frac{1}{3}x = 1x\) or simply \x\:\[ x + 5 = 17 \]
03

Isolate the Variable

Subtract 5 from both sides to solve for \x\:\[ x + 5 - 5 = 17 - 5 \]This simplifies to:\[ x = 12 \]

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Key Concepts

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

Combining Like Terms
When solving linear equations, one of the first steps is often combining like terms. This means adding or subtracting terms that have the same variable. In the given exercise, we combined \(\frac{2}{3}x\) and \(鈭抃frac{1}{3}x\). To achieve this, we moved all terms with the variable \x\ to one side of the equation.

Here鈥檚 a quick way to identify like terms:
  • They have the same variable.
  • They have the same exponent.
By adding \(\frac{1}{3}x\) to both sides, you get:

$$\frac{2}{3}x + \frac{1}{3}x + 5 = 17$$

After combining:
$$x + 5 = 17$$Combining like terms simplifies the equation and makes it easier to move on to the next steps.
Isolating the Variable
The goal of solving an equation is to find the value of the variable. To do this, you need to isolate the variable on one side of the equation. In our example, the equation was simplified to \(x + 5 = 17\). To isolate \x\, we eliminate the constant term on the same side as \x\.

Add or subtract constants as needed:
  • Here we subtract 5 from both sides.
  • Always perform the same operation on both sides to maintain equality.
\(x + 5 - 5 = 17 - 5\) simplifies to:

\(x = 12\)

Now, the variable is isolated, and you have found its value.
Simplification of Equations
Simplifying equations involves reducing them to their simplest form. After combining like terms and isolating the variable, we have \(x = 12\). This is the simplest form of the equation, and it gives us the solution directly.

Some steps to keep in mind:
  • Always perform operations carefully to avoid errors.
  • Make sure each equation is balanced by performing the same operation on both sides.
  • Continue simplifying until no further reduction is possible.
By following these steps, you can tackle more complex equations systematically.

A simplified equation usually means you have obtained the solution or are very close to it. You only stop simplifying when you have isolated the variable and have it equated to a constant.

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