Question

Solve the equation. Check your solution.
4gplus2​(negative 5plus2​g)equals1minusg
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Part 1
gequals
  
enter your response here ​(Type an integer or a simplified​ fraction.)

Answer 1

To solve the equation \( 4g + 2(-5 + 2g) = 1 - g \), let's first distribute and simplify:

1. Distributing \( 2 \) in the term \( 2(-5 + 2g) \):
\[
2(-5) + 2(2g) = -10 + 4g
\]
Now substitute this back into the equation:
\[
4g - 10 + 4g = 1 - g
\]

2. Combine like terms on the left side:
\[
4g + 4g - 10 = 1 - g
\]
which simplifies to:
\[
8g - 10 = 1 - g
\]

3. Now, let's get all the \( g \) terms on one side and constant terms on the other side. Add \( g \) to both sides:
\[
8g + g - 10 = 1
\]
which simplifies to:
\[
9g - 10 = 1
\]

4. Next, add \( 10 \) to both sides:
\[
9g = 11
\]

5. Finally, divide both sides by \( 9 \):
\[
g = \frac{11}{9}
\]

Thus, the solution for \( g \) is:
\[
g = \frac{11}{9}
\]

### Check the Solution
Substituting \( g = \frac{11}{9} \) back into the original equation for verification:

1. Calculate \( 4g + 2(-5 + 2g) \):
\[
4\left(\frac{11}{9}\right) + 2\left(-5 + 2\left(\frac{11}{9}\right)\right)
\]
First, \( 4\left(\frac{11}{9}\right) = \frac{44}{9} \).

Next, calculate \( 2\left(-5 + 2\left(\frac{11}{9}\right)\right) \):
\[
2\left(-5 + \frac{22}{9}\right) = 2\left(-\frac{45}{9} + \frac{22}{9}\right) = 2\left(-\frac{23}{9}\right) = -\frac{46}{9}
\]

Now combine:
\[
\frac{44}{9} - \frac{46}{9} = \frac{44 - 46}{9} = -\frac{2}{9}
\]

2. Calculate the right side:
\[
1 - g = 1 - \frac{11}{9} = \frac{9}{9} - \frac{11}{9} = -\frac{2}{9}
\]

Both sides are equal:
\[
-\frac{2}{9} = -\frac{2}{9}
\]

Thus, the solution \( g = \frac{11}{9} \) is verified.