To find solutions for the linear equation \(2x - 3(y - 2) = 1\), we can start by rewriting it in slope-intercept form.
Step 1: Simplifying the Equation
The given equation is:
\[ 2x - 3(y - 2) = 1 \]
Distributing the \(-3\):
\[ 2x - 3y + 6 = 1 \]
Subtracting 6 from both sides:
\[ 2x - 3y = -5 \]
Now, we can express \(y\) in terms of \(x\):
\[ -3y = -2x - 5 \]
Dividing by \(-3\):
\[ y = \frac{2}{3}x + \frac{5}{3} \]
Step 2: Finding Solutions
Now that we have the equation in slope-intercept form, we can find specific values for \(x\) and determine corresponding \(y\) values.
Solutions:
Here are four pairs of \((x, y)\) solutions:
- For \(x = 0\):
- \(y = \frac{2}{3}(0) + \frac{5}{3} = \frac{5}{3}\)
- Solution: \((0, \frac{5}{3})\)
- For \(x = 3\):
- \(y = \frac{2}{3}(3) + \frac{5}{3} = 2 + \frac{5}{3} = \frac{11}{3}\)
- Solution: \((3, \frac{11}{3})\)
- For \(x = 6\):
- \(y = \frac{2}{3}(6) + \frac{5}{3} = 4 + \frac{5}{3} = \frac{17}{3}\)
- Solution: \((6, \frac{17}{3})\)
- For \(x = -3\):
- \(y = \frac{2}{3}(-3) + \frac{5}{3} = -2 + \frac{5}{3} = \frac{-6 + 5}{3} = \frac{-1}{3}\)
- Solution: \((-3, \frac{-1}{3})\)
Summary of Solutions:
- \((0, \frac{5}{3})\)
- \((3, \frac{11}{3})\)
- \((6, \frac{17}{3})\)
- \((-3, \frac{-1}{3})\)
These solutions represent points that lie on the line defined by the equation.