Solve the equation. Express numbers as integers or simplified fractions.
[tex]$8 z-4=12 z+9$[/tex]
The solution set is $\square$.
Answer (2)
The solution to the equation 8 z − 4 = 12 z + 9 is z = − 4 13 . After isolating the variable step by step, we find the simplified fraction that represents z . Hence, the solution set is { − 4 13 } .
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Subtract 8 z from both sides: − 4 = 4 z + 9 .
Subtract 9 from both sides: − 13 = 4 z .
Divide both sides by 4: z = − 4 13 .
The solution set is − 4 13 .
Explanation
Understanding the Problem We are given the equation 8 z − 4 = 12 z + 9 and we want to solve for z . This involves isolating z on one side of the equation.
Subtracting 8z from both sides First, let's subtract 8 z from both sides of the equation to get the z terms on one side:
8 z − 4 − 8 z = 12 z + 9 − 8 z
This simplifies to:
− 4 = 4 z + 9
Subtracting 9 from both sides Next, we subtract 9 from both sides to isolate the term with z :
− 4 − 9 = 4 z + 9 − 9
This simplifies to:
− 13 = 4 z
Dividing by 4 Now, we divide both sides by 4 to solve for z :
4 − 13 = 4 4 z
This simplifies to:
z = − 4 13
Final Answer Therefore, the solution set is { 4 − 13 } .
Examples
Imagine you're comparing two different phone plans. Plan A costs $4 per month but charges $8 per gigabyte of data, while Plan B costs $9 per month but charges $12 per gigabyte. The equation 8 z − 4 = 12 z + 9 is similar to finding out how many gigabytes ( z ) you'd need to use for the two plans to cost the same. Solving such equations helps you make informed decisions about which plan is more economical for your data usage.
