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Chapter 2: Problem 65

Solve each equation, and check your solution. $$ -6(2 b+1)+(13 b-7)=0 $$

Short Answer

Expert verified
The solution is b = 13.

Step by step solution

01

- Distribute the -6

Apply the distributive property to \-6(2b+1). This means multiplying -6 by each term inside the parentheses: \[ -6 \times 2b + (-6 \times 1) = -12b - 6 \] Replace \-6(2b+1) with \-12b-6 in the original equation.
02

- Combine like terms

Now rewrite the equation with the distributed terms: \[ -12b - 6 + 13b - 7 = 0 \]Next, combine like terms, which are \-12b and \13b: \[ b - 13 = 0 \]
03

- Solve for b

Isolate the variable \b by adding 13 to both sides of the equation: \[ b - 13 + 13 = 0 + 13 \] This simplifies to: \[ b = 13 \]
04

- Check the solution

Substitute \b = 13 back into the original equation to verify: \[ -6(2(13) + 1) + (13(13) - 7) = 0 \] Simplify inside the parentheses first: \[ -6(26 + 1) + (169 - 7) = 0 \] This becomes: \[ -6(27) + 162 = 0 \] Calculate the products: \[ -162 + 162 = 0 \] The equation is verified as true.

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Key Concepts

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

Distributive Property
The distributive property is a key concept in solving equations. It helps to remove parentheses by distributing a multiplication over addition or subtraction inside the parentheses. In the equation \-6(2b + 1), distribute \-6 to both \2b and \1: \[ -6 \times 2b + (-6 \times 1) = -12b - 6 \] By doing this, we get rid of the parentheses, making the equation easier to work with. Rewriting the equation without parentheses helps clarify the next steps.
Combining Like Terms
Once you have used the distributive property, it is time to combine like terms. This involves adding or subtracting the coefficients of the same variables. For example, in the equation: \[ -12b - 6 + 13b - 7 = 0 \] Combine the terms with \b: \[ -12b + 13b \] This gives: \[ b - 13 \] Combining like terms simplifies the equation, making it more straightforward to solve. Always make sure to group similar terms for clarity.
Isolating the Variable
Isolating the variable means rearranging the equation to get the variable on one side by itself. In the simplified equation: \[ b - 13 = 0 \] You can isolate \b by adding \13 to both sides: \[ b - 13 + 13 = 0 + 13 \] This results in: \[ b = 13 \] Making the variable standalone allows you to easily see its value. It's crucial to perform the same operation on both sides to maintain the equation's balance.
Checking Solutions
Always check your solution by substituting it back into the original equation. This ensures the solution is correct. If \b = 13, substitute \b in: \[ -6(2(13) + 1) + (13(13) - 7) = 0 \] Simplify inside the parentheses: \[ -6(26 + 1) + (169 - 7) = 0 \] This results in: \[ -6(27) + 162 = 0 \] Finally, calculate: \[ -162 + 162 = 0 \] Since both sides are equal, \b = 13 is correct. Checking helps catch any possible errors made during solving.

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