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Chapter 2: Problem 193

Solve the following equations with variables on both sides. $$5 z=39-8 z$$

Short Answer

Expert verified
The solution is \(z = 3\).

Step by step solution

01

Move Variables to One Side

Add \(8z\) to both sides of the equation to move all variable terms to one side. This gives: \[5z + 8z = 39 - 8z + 8z\] Simplifying this, we get: \[13z = 39\]
02

Isolate the Variable

Divide both sides of the equation by 13 to solve for \(z\): \[z = \frac{39}{13}\] Simplify the right-hand side to get: \[z = 3\]
03

Solution Verification

Substitute \(z = 3\) back into the original equation to verify the solution: \[5(3) = 39 - 8(3)\] Simplifying both sides, we get: \[15 = 39 - 24\] \[15 = 15\] Since both sides are equal, the solution \(z = 3\) is correct.

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Key Concepts

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

Solving Linear Equations with Variables on Both Sides
When solving linear equations, you often encounter equations with variables on both sides. This can seem tricky at first, but with a few simple steps, you can manage it. The main goal is to get all terms with the variable on one side of the equation and the constants on the other side.
In the provided example, the equation is \[5z = 39 - 8z\]
Here's how you can approach it:
  • Identify the terms with the variable. In this case, they are \(5z\) and \(-8z\).
  • Decide which side of the equation you want all variable terms to be on. Usually, it's easiest to move variables to the side where they will remain positive.
  • Perform the same operation (addition or subtraction) on both sides to move the variable terms together.
This strategy makes sure you maintain the balance of the equation while simplifying it, leading you to an understandable and easy-to-solve expression.
Isolate the Variable
Once you have all the variable terms on one side and constants on the other, your next goal is to isolate the variable. This means getting the variable by itself on one side of the equation.
For the equation \(13z = 39\):
  • Identify the coefficient of the variable. Here, it's 13, which is multiplying \(z\).
  • To isolate \(z\), you need to perform the inverse operation. Since 13 is being multiplied by \(z\), you should divide both sides by 13.
So, divide both sides:
\[z = \frac{39}{13}\]
Once you simplify, you get \[z = 3\]. Now, \(z\) is isolated, and you have your solution!
Isolation is crucial because it allows you to find the exact value of the variable in question. Always remember to perform the same operation on both sides to keep the equation balanced.
Solution Verification
Finding a solution is great, but verifying it ensures your solution is correct. Here's how you can check your work:
  • Substitute your solution back into the original equation to replace the variable with the found value. In our case, substitute \(z = 3\) back into \(5z = 39 - 8z\).
  • Simplify both sides of the equation with this value to see if they are equal.
So, substitute and simplify:
\[5(3) = 39 - 8(3)\]
\[15 = 39 - 24\]
\[15 = 15\]
Since both sides are equal, the solution \(z = 3\) is indeed correct. Always perform this verification step to confirm that your solution satisfies the original equation. It鈥檚 a good habit and ensures you have not made any errors during your calculations.

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