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Chapter 4: Problem 9

Simplify and solve each equation. $$ 4-2(-5 a-10)=30 $$

Short Answer

Expert verified
a = \frac{3}{5}

Step by step solution

01

Distribute the Negative Sign

Distribute the -2 inside the parentheses to both -5a and -10. The equation becomes: \[ 4 - 2(-5a - 10) = 30 \]Which simplifies to:\[ 4 + 10a + 20 = 30 \]
02

Combine Like Terms

Combine the constant terms on the left-hand side of the equation. So now the equation becomes:\[ 4 + 20 + 10a = 30 \]This simplifies to:\[ 24 + 10a = 30 \]
03

Isolate the Variable Term

Subtract 24 from both sides of the equation to isolate the term with the variable:\[ 24 + 10a - 24 = 30 - 24 \]This simplifies to:\[ 10a = 6 \]
04

Solve for the Variable

Divide both sides by 10 to solve for a:\[ \frac{10a}{10} = \frac{6}{10} \]This simplifies to:\[ a = \frac{3}{5} \]

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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 helps simplify equations by distributing multiplication over addition or subtraction. In the given problem, the equation is:

$$4 - 2(-5a - 10) = 30$$

Here, we need to distribute \-2 inside the parentheses. This means multiplying each term inside by \-2:

$$ -2 \cdot (-5a) + (-2) \cdot (-10) $$

Which simplifies to:

$$ 10a + 20 $$

So, the equation becomes:

$$ 4 + 10a + 20 = 30 $$

Notice how the distributive property allowed us to remove the parentheses and simplified our expression. This strategy is particularly useful in more complex equations.
Combining Like Terms
Combining like terms involves adding or subtracting terms with the same variable to simplify an equation. In our problem, after applying the distributive property, we have:

$$ 4 + 10a + 20 = 30 $$

To combine like terms, we look for constants and terms with the variable a. Here, 4 and 20 are constants, and they can be added together:

$$ 4 + 20 = 24 $$
So, the equation simplifies to:

$$ 24 + 10a = 30 $$

Combining like terms makes solving the equation more straightforward, as it reduces the number of terms we need to work with.
Isolating the Variable
Isolating the variable means arranging the equation so that the variable (in this case, a) is on one side by itself. From the previous step, we have:

$$ 24 + 10a = 30 $$

First, we need to move the constant term (24) to the other side by subtracting it from both sides:

$$ 24 + 10a - 24 = 30 - 24 $$

Which simplifies to:

$$ 10a = 6 $$

Now, to solve for a, we divide both sides by 10:

$$ \frac{10a}{10} = \frac{6}{10} $$

So, we get:

$$ a = \frac{3}{5} $$

Isolating the variable helps to clearly find the value of the unknown variable that makes the equation true. It is the final step after simplifying through distribution and combining like terms.

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Most popular questions from this chapter

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