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Chapter 2: Problem 39

Solve each equation. $$6 a-5(a-2)+9=-11$$

Short Answer

Expert verified
a = -30

Step by step solution

01

Distribute the Constants

Distribute the \(-5\) across the terms inside the parenthesis \((-5(a-2))\). The equation becomes: \(6a - 5a + 10 + 9 = -11\)
02

Combine Like Terms

Combine the \(a\) terms and the constants on the left-hand side: \((6a - 5a) + (10 + 9) = -11\) The equation becomes: \(a + 19 = -11\)
03

Isolate the Variable

Subtract \(19\) from both sides to isolate \(a\): \(a + 19 - 19 = -11 - 19\) The equation simplifies to: \(a = -30\)

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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 core concept in algebra that allows you to simplify expressions by distributing a constant across terms inside parentheses. To apply the distributive property, you multiply the constant by each term within the parentheses. For example, in the equation 6a - 5(a - 2) + 9 = -11, you distribute -5 across a - 2. This means: -5 * a + (-5) * (-2). So, -5 * a = -5a and -5 * (-2) = 10. The equation then transforms to 6a - 5a + 10 + 9 = -11.
Combining Like Terms
Combining like terms is the next key step. 'Like terms' are terms that have the same variables raised to the same power. For instance, 6a and -5a are like terms because both contain the variable 'a'. Similarly, the constants 10 and 9 are like terms. In the equation 6a - 5a + 10 + 9 = -11, you combine 6a and -5a to get a. Then, you combine 10 and 9 to get 19. So, the simplified equation is a + 19 = -11. This makes the equation easier to solve.
Isolating Variables
The last important step in solving the equation is isolating the variable. This usually means getting the variable (in this case, 'a') by itself on one side of the equation. In a + 19 = -11, you isolate a by subtracting 19 from both sides of the equation. This is done to balance the equation without changing its equality. So, a + 19 - 19 = -11 - 19 simplifies to a = -30. Now, the variable is isolated, and the solution to 6a - 5(a-2) + 9 = -11 is a = -30.

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