Write the equation of a line through (3, 1), parallel to y = 2/3 x + 1. Write the equation of a line through (-3, 5), parallel to y = -x.

Understand the Problem

The questions are asking for the equations of lines that pass through specific points and are parallel to given lines. This involves using the slope of the given lines to formulate the equations.

Answer

1. $y = \frac{2}{3}x - 1$ 2. $y = -x + 2$

Steps to Solve

Identify the slope of the given line

For Question 1, the equation of the line is given as $y = \frac{2}{3}x + 1$. The slope is $\frac{2}{3}$.

For Question 2, the equation is $y = -x$. The slope is $-1$.

Use the point-slope form of a line

The point-slope form of the equation of a line is given by:

$$ y - y_1 = m(x - x_1) $$

Where $(x_1, y_1)$ is the point the line goes through, and $m$ is the slope.

Apply the point-slope formula to each question

For Question 1, the point is $(3, 1)$ and the slope is $\frac{2}{3}$:

$$ y - 1 = \frac{2}{3}(x - 3) $$

For Question 2, the point is $(-3, 5)$ and the slope is $-1$:

$$ y - 5 = -1(x + 3) $$

Rearranging the equations

For Question 1, distribute and simplify:

$$ y - 1 = \frac{2}{3}x - 2 $$

So,

$$ y = \frac{2}{3}x - 1 $$

For Question 2, distribute and simplify:

$$ y - 5 = -x - 3 $$

So,

$$ y = -x + 2 $$

The equation for Question 1 is $y = \frac{2}{3}x - 1$.
The equation for Question 2 is $y = -x + 2$.

More Information

For Question 1, the line through the point (3, 1) is parallel to the line with slope $\frac{2}{3}$.

For Question 2, the line through (-3, 5) is parallel to the line with slope $-1$.

Tips

Forgetting to use the correct slope: Ensure that you take the slope from the given line associated with the correct question.
Incorrectly substituting point coordinates: Double-check that you substitute the coordinates of the point into the point-slope formula accurately.

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