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Chapter 3: Problem 68

Solve the equation. $$-3+11=-2 z+3(z-2)$$

Short Answer

Expert verified
z = 14

Step by step solution

01

- Simplify Both Sides of the Equation

Combine like terms on both sides of the equation. Given: \[ -3 + 11 = -2z + 3(z - 2) \] First, simplify the left-hand side: \[ -3 + 11 = 8 \] Next, simplify the right-hand side by distributing the 3: \[ -2z + 3(z - 2) = -2z + 3z - 6 = z - 6 \]
02

- Set the Simplified Equations Equal to Each Other

Now we have: \[ 8 = z - 6 \] This is a simpler equation where we can isolate the variable easily.
03

- Solve for z

Add 6 to both sides to isolate the variable z: \[ 8 + 6 = z - 6 + 6 \] Simplify: \[ 14 = z \] Therefore: \[ z = 14 \]

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Key Concepts

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

Simplifying Equations
Simplifying equations makes them easier to solve. To simplify, combine like terms, which are terms that have the same variable raised to the same power, or constants. For example, given -3 + 11 = -2z + 3(z - 2) First, simplify each side separately. Start with the left-hand side: -3 + 11 = 8 Now, simplify the right-hand side. Apply the distributive property: -2z + 3(z - 2) = -2z + 3 * z - 3 * 2 = -2z + 3z - 6 Combine the terms containing z: -2z + 3z = z So, it simplifies to: z - 6 After simplifying both sides, we get: 8 = z - 6
Simplifying helps you to focus on the core expression and makes solving the equation straightforward.
Isolating Variables
To solve an equation, you need to isolate the variable. Isolating the variable means getting the variable by itself on one side of the equation. Start with the equation from the first step: 8 = z - 6 To isolate z, we need to undo the subtraction of 6. We do this by adding 6 to both sides: 8 + 6 = z - 6 + 6 Adding the values gives: 14 = z Now we have z by itself: z = 14
Always perform the same operation on both sides of the equation to keep it balanced. By the end, z or any variable should be isolated, giving you the solution.
Distributive Property
The distributive property is a handy tool for simplifying equations, especially when dealing with parentheses. The property states: a(b + c) = ab + ac This means you multiply each term inside the parentheses by the term outside. For the equation -2z + 3(z - 2) apply the distributive property: 3(z - 2) = 3 * z - 3 * 2 = 3z - 6 So, the equation becomes: -2z + 3z - 6 Combine like terms: -2z + 3z = z Therefore, -2z + 3(z - 2) = z - 6
Understanding and applying the distributive property helps to eliminate parentheses and simplifies the equation, making the next steps easier to handle.

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