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Chapter 6: Problem 15

Solve each equation, and check the solutions. $$ \frac{3 x}{5}-6=x $$

Short Answer

Expert verified
x = -15

Step by step solution

01

- Eliminate the fraction

Multiply both sides of the equation by 5 to eliminate the fraction:\[5 \times \frac{3x}{5} - 5 \times 6 = 5 \times x\]This simplifies to:\[3x - 30 = 5x\]
02

- Isolate the variable term

Subtract 3x from both sides of the equation to isolate the variable term on one side:\[3x - 30 - 3x = 5x - 3x\]This simplifies to:\[-30 = 2x\]
03

- Solve for x

Divide both sides by 2 to solve for x:\[\frac{-30}{2} = \frac{2x}{2}\]This simplifies to:\[x = -15\]
04

- Check the solution

Substitute \(x = -15\) back into the original equation to verify the solution:\[\frac{3(-15)}{5} - 6 = -15\]Simplify:\[\frac{-45}{5} - 6 = -15\]\[-9 - 6 = -15\]\[-15 = -15\]The solution \(x = -15\) satisfies the original equation.

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Key Concepts

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

eliminating fractions
When solving linear equations, fractions can make the process difficult. It's usually best to eliminate fractions early.
To do this, you can multiply every term of the equation by the denominator of the fraction. This makes the equation easier to work with.
In the problem \[ \frac{3x}{5} - 6 = x \], multiplying each term by 5 helps remove the fraction:
\[ 5 \times \frac{3x}{5} - 5 \times 6 = 5 \times x \]
This leaves you with:
\[ 3x - 30 = 5x \]

isolating variables
Once fractions are eliminated, you need to isolate the variable on one side of the equation.
This means you want all terms with the variable on one side and all constant terms on the other.
In our equation, we have \[ 3x - 30 = 5x \].
Subtract 3x from both sides to isolate x:
\[ 3x - 30 - 3x = 5x - 3x \]
This simplifies to:
\[ -30 = 2x \]

solving for x
After isolating the variable, the next step is to solve for it.
Now we have the simplified equation from before \[ -30 = 2x \].
To solve for x, divide both sides by the coefficient of x.
This looks like:
\[ \frac{-30}{2} = \frac{2x}{2} \]
Which simplifies to:
\[ x = -15 \]
So, the value of x is -15.
checking solutions
The last step is always to check your solution.
This ensures your answer is correct and satisfies the original equation.
Substitute x back into the original equation \[ \frac{3(-15)}{5} - 6 = -15 \].
First, simplify within the fraction:
\[ \frac{-45}{5} - 6 = -15 \]
Then continue simplifying:
\[ -9 - 6 = -15 \]
Finally, you'll see that \[ -15 = -15 \].
This confirms that our solution x = -15 is indeed correct. Always checking your result helps to ensure you're on the right track and verifies your steps.

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Most popular questions from this chapter

Let \(P\), \(Q\), and \(R\) be rational expressions defined as follows. $$P=\frac{6}{x+3}, \quad Q=\frac{5}{x+1}, \quad R=\frac{4 x}{x^{2}+4 x+3}$$ Find the values for which each expression is undefined. Write answers using the symbol \(\neq\) (a) \(P\) (b) \(Q\) (c) \(R\)

Simplify each complex fraction. Use either method. $$ \frac{\frac{1}{x}+x}{\frac{x^{2}+1}{8}} $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{5}{x^{2} y}-\frac{2}{x y^{2}}}{\frac{3}{x^{2} y^{2}}+\frac{4}{x y}} $$

Find the slope of the line that passes through each pair of points. This will involve simplifying complex fractions. $$ \left(-\frac{2}{9}, \frac{5}{18}\right) \text { and }\left(\frac{1}{18},-\frac{5}{9}\right) $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{1}{a+1}}{\frac{2}{a^{2}-1}} $$
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