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Chapter 2: Problem 46

Solve each equation, and check your solution. $$ 8 z=7 z $$

Short Answer

Expert verified
The solution to the equation is \(z = 0\).

Step by step solution

01

Isolate the variable

Begin by isolating the variable on one side of the equation. Subtract 7z from both sides to start the process.\[ 8z - 7z = 7z - 7z \]
02

Simplify the equation

Simplify the equation after performing the subtraction on both sides.\[ z = 0 \]
03

Check the solution

Substitute the solution back into the original equation to verify it. Substitute \(z = 0\) into \(8z = 7z\): \[ 8(0) = 7(0) \]Both sides equal 0, so the solution is verified.

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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 linear equations, the first crucial step is to isolate the variable. This means getting the variable you are solving for alone on one side of the equation.
This process helps to clearly see the value of the variable without any additional terms or coefficients.
In our example, we started with the equation: 8z = 7z.
To isolate the variable, we subtract 7z from both sides: 8z - 7z = 7z - 7z.
The subtraction leaves us with:
z = 0.
This step is foundational because it simplifies the task of finding what value the variable represents.
It鈥檚 like clearing the fog around the variable so you can see it clearly.
Simplifying Equations
After isolating the variable, the next step is to simplify the equation.
Simplifying equations involves combining like terms and reducing any complex expressions.
In the equation we are working on, once we isolated the variable, we had: 8z - 7z = 0.
Simplifying it by performing the subtraction results in:
z = 0.
This means we now have a clean, straightforward expression for the variable.
Simplifying equations makes them easier to understand and solve.
It also reduces the risk of making mistakes in further steps.
Checking Solutions
The final step in solving linear equations is checking your solution.
Checking solutions helps verify that your solution is correct.
To do this, substitute the variable's value back into the original equation and see if both sides of the equation remain equal.
In our equation, substituting z = 0 back into 8z = 7z gives us:
8(0) = 7(0).
Since both sides equal zero, we confirm that our solution is correct.
This step is crucial as it ensures that no mistakes were made during the solving process.
It validates your hard work and boosts your confidence in handling similar problems efficiently.

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Most popular questions from this chapter

The euro is the common currency used by most European countries, including Italy. On August \(15,2009,\) the exchange rate between euros and U.S. dollars was 1 euro to \(\$ 1.4294\) Ashley went to Rome and exchanged her U.S. currency for euros, receiving 300 euros. How much in U.S. dollars did she exchange?

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