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Chapter 5: Problem 28

Solve and check. $$z-6=10$$

Short Answer

Expert verified
The solution to the equation \( z-6 = 10 \) is \( z = 16 \)

Step by step solution

01

Identify and isolate the variable on one side of the equation

The equation, \(z-6=10\), has the number -6 added to z. To isolate z, add 6 to both sides of the equation to get \(z = 10 + 6\)
02

Simplify the equation

Combine the values on the right side of the equation to get \( z = 16 \)
03

Check the solution

To check if \( z= 16 \) is the solution to the equation, substitute 16 for z in the original equation: \(16 - 6 = 10\). Since the left side of the equation (10) is equal to the right side of the equation (10), the solution for z is indeed 16.

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Key Concepts

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

Isolate the Variable
When solving a linear equation, isolating the variable is often the first step. This means getting the variable you are trying to solve for -- in this case, \( z \) -- by itself on one side of the equation. Think of it like cleaning up a messy desk; you want just one item in a particular spot without any clutter. Here鈥檚 how you do it:
  • Look at the equation and find the term associated with the variable.
  • Identify any numbers or operations being conducted on this variable.
  • In our example, the equation is \( z - 6 = 10 \). The number \(-6\) is subtracting from \( z \).
To isolate \( z \), you have to undo this subtraction. You can do this by adding 6 to both sides of the equation. When you add or subtract the same value from both sides, you maintain balance. This results in \( z = 16 \), letting you know what the variable represents.
Substitution Method
The substitution method involves plugging the solution you've obtained back into the original equation. This helps verify that the solution is correct and that the variable was isolated correctly. After you isolate the variable and find that \( z = 16 \), you use substitution to make sure this value works:
  • Replace \( z \) in the original equation with \( 16 \).
  • The original equation, \( z - 6 = 10 \), becomes \( 16 - 6 = 10 \).
This substitution checks the value of \( z \): - If after substitution, both sides of the equation are equal (meaning the calculation holds), then the solution you've found is correct. In this case, 10 indeed equals 10, so \( z = 16 \) is the right solution.
Check Solution
Checking the solution is a vital part of solving an equation as it confirms whether the result is accurate. It鈥檚 like proofreading an essay to ensure there are no errors.Here's how you check:
  • Take your value from the isolated variable, which is \( 16 \), and put it back into the original equation to see if it satisfies both sides.
  • For the equation \( z - 6 = 10 \), substituting \( z \) with \( 16 \) gives \( 16 - 6 = 10 \).
Confirm both sides balance:- The left side of your equation equals 10, the same as the right side.Ensuring the left and right sides of the equation match tells you that \( z = 16 \) is definitely the correct solution. This kind of double-checking is crucial, especially in more complex equations.

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