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Chapter 2: Problem 19

Solve and check. Label any contradictions or identities. $$-6 z-18=-132$$

Short Answer

Expert verified
The solution is \(z = 19\). It checks out since \(-6(19) - 18 = -132\).

Step by step solution

01

Isolate the variable term

Add 18 to both sides of the equation to isolate the term with the variable on one side. \[-6z - 18 + 18 = -132 + 18\] This simplifies to: \[-6z = -114\]
02

Solve for the variable

Divide both sides of the equation by -6 to solve for the variable \(z\). \[\frac{-6z}{-6} = \frac{-114}{-6}\] This simplifies to: \[z = 19\]
03

Check the solution

Substitute \(z = 19\) back into the original equation to verify the solution. \[-6(19) - 18 = -132\] Calculate: \[-114 -18 = -132\] Since both sides of the equation are equal, 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
To solve any linear equation, the first step is often isolating the variable. This means you want to get the variable by itself on one side of the equation. In the given problem, we start with the equation: \[-6z - 18 = -132\].
To isolate the term with the variable, we need to get rid of the constant term on that side.
We achieve this by performing the inverse operation. Here, since the term -18 is being subtracted, we add 18 to both sides of the equation.
This simplifies to: \[-6z - 18 + 18 = -132 + 18\], which becomes: \[-6z = -114\].
By isolating the variable term, we are making the equation simpler and getting closer to finding the value of z.
Variables
Understanding variables is crucial in solving linear equations. A variable is a symbol, usually a letter, that represents an unknown value in a mathematical equation.
In the given problem, z is our variable. At the beginning, z is multiplied by -6 and then subtracted by 18.
Our goal is to find out what value of z makes the equation true.
Once we've isolated the term with the variable, we proceed by solving for z. In this case, we divide both sides by -6: \[ \frac{-6z}{-6} = \frac{-114}{-6}\]. This gives us: \[ z = 19\].
Thus, we have determined that when z is 19, the original equation holds true.
Verification of Solutions
After finding a potential solution, it's always important to verify that our solution is correct. Verification means substituting the value we found back into the original equation to ensure both sides are equal.
For our example, we substitute z = 19 back into the original equation: \[-6(19) - 18 = -132\].
We then perform the calculation: \[-114 - 18 = -132\]. Since this statement is true (both sides of the equation are indeed equal), we have verified that our solution, z = 19, is correct.
Always remember that verification is a crucial step because it ensures that we haven鈥檛 made any mistakes in our calculations.

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