Solve each equation: 3y - 6 / 9 = 4 - 2y / 3 and 0.3(x - 10) - 1.8 = 2.7x.
Understand the Problem
The question is asking to solve two algebraic equations. The first equation involves y and the second equation involves x, requiring an understanding of algebraic manipulation to find the values of these variables.
Answer
The solution for \(y\) is \(y = 2\) and for \(x\) is \(x = -2\).
Answer for screen readers
The solutions are ( y = 2 ) and ( x = -2 ).
Steps to Solve
- Solve the first equation for (y)
Start with the equation:
$$ \frac{3y - 6}{9} = \frac{4 - 2y}{3} $$
Multiply both sides by 9 to eliminate the denominator:
$$ 3y - 6 = 3(4 - 2y) $$
Expand the right side:
$$ 3y - 6 = 12 - 6y $$
- Combine like terms
Add (6y) to both sides:
$$ 3y + 6y = 12 + 6 $$
This simplifies to:
$$ 9y = 18 $$
- Solve for (y)
Divide both sides by 9:
$$ y = 2 $$
- Solve the second equation for (x)
Start with the equation:
$$ 0.3(x - 10) - 1.8 = 2.7x $$
Distribute (0.3):
$$ 0.3x - 3 - 1.8 = 2.7x $$
Combine the constants on the left side:
$$ 0.3x - 4.8 = 2.7x $$
- Isolate (x)
Subtract (0.3x) from both sides:
$$ -4.8 = 2.7x - 0.3x $$
This simplifies to:
$$ -4.8 = 2.4x $$
- Solve for (x)
Divide both sides by 2.4:
$$ x = \frac{-4.8}{2.4} = -2 $$
The solutions are ( y = 2 ) and ( x = -2 ).
More Information
The process involved solving two different algebraic equations. The first equation was solved by simplifying and isolating (y), while the second equation required distribution and isolating (x).
Tips
- Misapplying distribution: A common error is incorrectly distributing factors, especially in the second equation.
- Ignoring sign when moving terms: Ensure to keep track of signs when adding and subtracting terms across the equation.
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