Practice Solving Multistep Equations with Distribution

1. Solve for \(p\).
   

   \(10p-3 = 2(12+4p) -7\)

2. Solve for \(g\).
   

   \(9g = 3(-4+5g) \)

3. Solve for \(e\).
   

   \(7(2e-1) - 3 = 6+6e \)

4. Solve for \(m\).
   \(m=-(4+m)+2 \)

Source: Khan Academy, https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:solve-equations-inequalities/x2f8bb11595b61c86:linear-equations-parentheses/e/multistep_equations_with_distribution
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1. We need to manipulate the equation to get \(p\) by itself.

   \(10 p-3  = 2(12+4 p)-7\)
   \(10 p-3  = 24+8 p-7 \)	Distribute
   \(10 p-3  = 17+8p\)	Combine like terms
   \(10 p-3-8 p = 17+8 p-8p \)	Subtract \(8p\) from each side
   \(2 p-3  = 17\)	Combine like terms
   \(2 p-3+3  = 17+3\)	Add 3 to each side
   \(2 p  = 20 \)	Combine like terms
   \(\frac{2 p}{2}  = \frac{20}{2}\)	Divide each side by 2
   \(p  = 10\)	Simplify

   The answer:

   \(p=10\)

2. We need to manipulate the equation to get \(g\) by itself.

   \(9g  = 3(-4+5g) \)
   \(9g = -12+15g  \)	Distribute
   \(9g-15g  = -12+15g-15g   \)	Subtract \(15g\) from each side
   \(-6g  = -12 \)	Combine like terms
   \(\frac{-6g}{-6} = \frac{-12}{-6} \)	Divide each side by -6
   \(g = 2 \)	Simplify

   
   

   The answer:

   \(g=2\)

3. We need to manipulate the equation to get \(e\) by itself.

   \(7(2e-1) - 3 = 6+6e \)
   \(14e-7 - 3 = 6+6e \)	Distribute
   \(14e-10 = 6+6e \)	Combine like terms
   \(14e-10-6e = 6+6e -6e\)	Subtract \(6e\) from each side
   \(8e-10 = 6\)	Combine like terms
   \(8e-10+10 = 6+10\)	Add 10 to each side
   \(8e = 16\)	Combine like terms
   \(\frac{8e}{8} = \frac{16}{8}\)	Divide each side by 8
   \(e=2\)	Simplify

   The answer:

   \(e=2\)

4. We need to manipulate the equation to get \(m\) by itself.

   \(m=-(4+m)+2 \)
   \(m=-4-m+2 \)	Distribute
   \(m=-2-m \)	Combine like terms
   \(m+m=-2-m+m \)	Add \(m\) to each side
   \(2m=-2 \)	Combine like terms
   \(\frac{2m}{2} = \frac{-2}{2}\)	Divide each side by 2
   \(m=-1\)	Simplify

   
   The answer:
   
   \(m=-1\)