Solve the following two equations for their variables: 1. 5y + 14 = -43 - 3y 2. 4(2a + 2) = 8(2 - 3a)
Understand the Problem
The question presents two algebraic equations. The first equation is 5y + 14 = -43 - 3y, and asks to solve for the variable 'y'. The second equation is 4(2a + 2) = 8(2 - 3a), and asks to solve for the variable 'a'. We need to isolate the variable in each equation to find its value.
Answer
$y = -\frac{57}{8}$ $a = \frac{1}{4}$
Answer for screen readers
$y = -\frac{57}{8}$ $a = \frac{1}{4}$
Steps to Solve
- Solve for y: Combine 'y' terms
Add $3y$ to both sides of the equation $5y + 14 = -43 - 3y$ $5y + 3y + 14 = -43 - 3y + 3y$ $8y + 14 = -43$
- Isolate the 'y' term
Subtract 14 from both sides of the equation $8y + 14 = -43$ $8y + 14 - 14 = -43 - 14$ $8y = -57$
- Solve for 'y'
Divide both sides of the equation $8y = -57$ by 8 $\frac{8y}{8} = \frac{-57}{8}$ $y = -\frac{57}{8}$
- Solve for a: Distribute constants
Distribute the constants on both sides of $4(2a + 2) = 8(2-3a)$ $8a + 8 = 16 - 24a$
- Combine 'a' terms
Add $24a$ to both sides of the equation $8a + 8 = 16 - 24a$ $8a + 24a + 8 = 16 - 24a + 24a$ $32a + 8 = 16$
- Isolate the 'a' term
Subtract 8 from both sides of the equation $32a + 8 = 16$ $32a + 8 - 8 = 16 - 8$ $32a = 8$
- Solve for 'a'
Divide both sides of the equation $32a = 8$ by 32 $\frac{32a}{32} = \frac{8}{32}$ $a = \frac{1}{4}$
$y = -\frac{57}{8}$ $a = \frac{1}{4}$
More Information
The solution for y is a negative improper fraction, while the solution for a is a positive fraction less than 1.
Tips
- Forgetting to distribute the constant to all terms within the parentheses.
- Making arithmetic errors when adding or subtracting.
- Incorrectly dividing to isolate the variable.
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