The problem asks to analyze a line given by its graph. (a) Find the y-intercept of the line. (b) Find the slope of the line. (c) Find the equation of the line in standard form. Also, we need to find the equation of the line that is perpendicular to the line $3x+2y=1$ and passes through the point $(-2, 1)$. The answer must be in slope-intercept form.

1. Problem Description

The problem asks to analyze a line given by its graph.

(a) Find the y-intercept of the line.

(b) Find the slope of the line.

(c) Find the equation of the line in standard form.

Also, we need to find the equation of the line that is perpendicular to the line $3x+2y=1$ and passes through the point $(-2, 1)$. The answer must be in slope-intercept form.

2. Solution Steps

3. Given the graph of the line.

(a) The y-intercept is the point where the line crosses the y-axis. From the graph, the y-intercept is at $y = -1$. So the y-intercept is $(0, -1)$.

(b) To find the slope of the line, we can choose two points on the line. From the graph, we can see the points $(-4, 3)$ and $(0, -1)$.

The formula for slope $m$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(-4, 3)$ and $(0, -1)$, we have:

$m = \frac{-1 - 3}{0 - (-4)} = \frac{-4}{4} = -1$

So the slope of the line is $-1$.

(c) The standard form of a linear equation is $Ax + By = C$, where A, B, and C are integers, and A is non-negative. We know the slope $m = -1$ and the y-intercept is $(0, -1)$.

Using the slope-intercept form, we have:

$y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

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

$y = -x - 1$

To convert this to standard form, we add $x$ to both sides:

$x + y = -1$

4. Find the equation of the line that is perpendicular to the line $3x + 2y = 1$ and passes through the point $(-2, 1)$.

First, we need to find the slope of the given line $3x + 2y = 1$. To do this, we can rewrite the equation in slope-intercept form:

$2y = -3x + 1$

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

The slope of this line is $-\frac{3}{2}$.

Since we want a line perpendicular to this one, the slope of the new line will be the negative reciprocal of $-\frac{3}{2}$, which is $\frac{2}{3}$.

Now we know the slope of the new line is $\frac{2}{3}$ and it passes through the point $(-2, 1)$. We can use the point-slope form to find the equation of the line:

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

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

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

$y - 1 = \frac{2}{3}x + \frac{4}{3}$

To get the equation in slope-intercept form, we add 1 to both sides:

$y = \frac{2}{3}x + \frac{4}{3} + 1$

$y = \frac{2}{3}x + \frac{4}{3} + \frac{3}{3}$

$y = \frac{2}{3}x + \frac{7}{3}$

3. Final Answer

3. (a) The y-intercept is $(0, -1)$ or $y = -1$.

(b) The slope is $-1$.

(c) The standard form is $x + y = -1$.

4. The equation of the line is $y = \frac{2}{3}x + \frac{7}{3}$.