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Chapter 3: Problem 7

Solve the equation. $$-8=3-2(z-4)+3 z$$

Short Answer

Expert verified
z = -19

Step by step solution

01

Distribute the -2

Distribute the -2 across the terms inside the parentheses. -8 = 3 - 2(z - 4) + 3z -8 = 3 - 2z + 8 + 3z
02

Combine like terms

Combine the constant terms and the z terms on the right-hand side. -8 = (3 + 8) + (-2z + 3z)-8 = 11 + z
03

Isolate the variable z

Subtract 11 from both sides to solve for z. -8 - 11 = z -19 = z

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Key Concepts

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

Distribution in Algebra
In algebra, distribution refers to the process of expanding an expression that involves parentheses by multiplying each term inside the parentheses by the factor outside. This is known as the distributive property.
Let's look at the example:
-8 = 3 - 2(z - 4) + 3z
If we distribute the -2 across the terms inside the parentheses, we get:
-8 = 3 - 2z + 8 + 3z
Here鈥檚 how it works:
  • Multiply -2 by z, giving us -2z
  • Multiply -2 by -4, giving us +8
So, the expression inside the parentheses is replaced with these calculations. This way, we handle all terms before combining any like terms.
Combining Like Terms
After distributing, the next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power.
In our example:
-8 = 3 - 2z + 8 + 3z
We need to combine the constant terms and the terms that include z. Constants are numbers without variables. In this case, the constants are 3 and 8. The terms with the variable z are -2z and 3z.
We combine these like terms as follows:
  • Combine 3 and 8 to get 11 (the constants)
  • Combine -2z and 3z to get z
This simplifies our equation to:
-8 = 11 + z
Isolating Variables
The final step to solve the equation is to isolate the variable. This means getting the variable by itself on one side of the equation.
In the simplified equation:
-8 = 11 + z
We need to get z alone. To do this, subtract 11 from both sides of the equation:
-8 - 11 = z
Which simplifies to:
-19 = z
This is the value of the variable z. By isolating the variable, we've solved the equation!

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