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Chapter 1: Problem 13

Solve each equation and check your answer $$3 x-5=0$$

Short Answer

Expert verified
x = \( \frac{5}{3} \)

Step by step solution

01

Isolate the variable term

To solve for the variable, you need to isolate the term that contains the variable. Here we have the equation: \[ 3x - 5 = 0 \]Add 5 to both sides of the equation to get: \[ 3x - 5 + 5 = 0 + 5 \] which simplifies to: \[ 3x = 5 \]
02

Solve for the variable

Now, solve for \(x\) by dividing both sides of the equation by 3: \[ \frac{3x}{3} = \frac{5}{3} \] This simplifies to: \[ x = \frac{5}{3} \]
03

Check your answer

Substitute \( x = \frac{5}{3} \) back into the original equation to verify the solution: \[ 3\left( \frac{5}{3} \right) - 5 = 0 \] Simplify the left-hand side: \[ 5 - 5 = 0 \] This is a true statement, confirming that the solution is correct.

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Key Concepts

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

Isolating the Variable
One of the first steps in solving linear equations involves isolating the variable. To isolate a variable means to get the variable by itself on one side of the equation.
Start with the given equation: \[ 3x - 5 = 0 \]
We need to remove the constant term (in this case, -5) from the side with the variable. We do this by adding 5 to both sides.
This gives: \[ 3x - 5 + 5 = 0 + 5 \]
As a result, we simplify it to: \[ 3x = 5 \]
Now, our equation is simpler and has the variable isolated, ready for the next steps.
Checking Solutions
After finding the value of the variable, it's important to verify that your solution is correct. Start with the original equation: \[ 3x - 5 = 0 \]
Substitute your solution, which is \[ x = \frac{5}{3} \]
This gives: \[ 3\left( \frac{5}{3} \right) - 5 = 0 \]
Next, simplify the left-hand side.
The term \[ 3\left( \frac{5}{3} \right) \] simplifies to 5.
So, we have: \[ 5 - 5 = 0 \]
Since both sides match after substitution, the solution \[ x = \frac{5}{3} \] is verified as correct. It's crucial to always check this step to avoid mistakes.
Simplifying Equations
Simplifying equations means making the problem easier to solve by reducing it to its simplest form.
Take the equation: \[ 3x - 5 = 0 \]
First, we need to simplify by getting rid of terms that do not include the variable.
Add 5 to both sides to eliminate the -5 term: \[ 3x - 5 + 5 = 0 + 5 \]
This results in a cleaner equation: \[ 3x = 5 \]
Finally, solve for x by dividing everything by 3:
Thus, \[ x = \frac{5}{3} \]
By simplifying, we make the equation more straightforward, facilitating the solving process.

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$$\text { Solve } \frac{x}{2}+\frac{1}{3}=\frac{x}{9}+\frac{1}{6}$$.
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