What is the equation of a line parallel to 2x - 3y = 3 that contains the point (3, -5)?
Understand the Problem
The question asks for the equation of a line that is parallel to the given line and passes through a specific point. To find the equation, we need to determine the slope of the given line and use it to derive the new equation that passes through the point (3, -5).
Answer
$y = \frac{2}{3}x - 7$
Answer for screen readers
The equation of the line is $y = \frac{2}{3}x - 7$.
Steps to Solve
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Convert the given line equation to slope-intercept form
The given line is $2x - 3y = 3$. We will solve for $y$ to find the slope.
Rearranging the equation: $$ -3y = -2x + 3 $$ Dividing by -3: $$ y = \frac{2}{3}x - 1 $$ The slope ($m$) of the given line is $\frac{2}{3}$.
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Use the point-slope form of a line
Since we want a line parallel to the given line, it will have the same slope. We will use the point-slope form, which is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point $(3, -5)$.
Plugging in the values: $$ y - (-5) = \frac{2}{3}(x - 3) $$
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Simplify the point-slope equation
Distributing the slope: $$ y + 5 = \frac{2}{3}x - 2 $$ Now, isolate $y$: $$ y = \frac{2}{3}x - 2 - 5 $$ Simplifying further: $$ y = \frac{2}{3}x - 7 $$
The equation of the line is $y = \frac{2}{3}x - 7$.
More Information
This equation represents a line that is parallel to the original line $2x - 3y = 3$ and passes through the point (3, -5). Since parallel lines have the same slope, we maintained the slope of $\frac{2}{3}$ while incorporating the new point.
Tips
- Misunderstanding parallel lines: A common mistake is to use the slope of the point instead of the slope of the existing line. Remember that parallel lines have equal slopes.
- Incorrectly manipulating the equation: Ensure you follow the algebraic steps carefully to avoid mistakes when isolating $y$.
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