The problem requires us to analyze the given graph of a line. (a) We need to find the y-intercept of the line. (b) We need to calculate the slope of the line. (c) We need to express the equation of the line in standard form. (4) We need to find the equation of the line that is perpendicular to the line $3x + 2y = 1$ and passes through the point $(-2, 1)$. We need to express our answer in slope-intercept form.

1. Problem Description

The problem requires us to analyze the given graph of a line.

(a) We need to find the y-intercept of the line.

(b) We need to calculate the slope of the line.

(c) We need to express the equation of the line in standard form.

(4) We need to find the equation of the line that is perpendicular to the line $3x + 2y = 1$ and passes through the point $(-2, 1)$. We need to express our answer in slope-intercept form.

2. Solution Steps

(a) Find the y-intercept of the line shown:

From the graph, the line intersects the y-axis at $y = -0.5$.

So, the y-intercept is $-0.5$.

(b) Find the slope of the line shown:

From the graph, we can identify two points on the line: $(-5, 3)$ and $(0, -0.5)$.

The slope $m$ is given by the formula:

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

Substituting the coordinates of the two points:

$m = \frac{-0.5 - 3}{0 - (-5)} = \frac{-3.5}{5} = -\frac{7}{10} = -0.7$

So, the slope of the line is $-0.7$.

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

We have the slope $m = -0.7$ and the y-intercept $b = -0.5$.

The slope-intercept form of the equation is $y = mx + b$.

$y = -0.7x - 0.5$

To convert to standard form $Ax + By = C$, we multiply by 10 to eliminate decimals:

$10y = -7x - 5$

$7x + 10y = -5$

So, the equation in standard form is $7x + 10y = -5$.

(4) Find the equation of the line that is perpendicular to the line $3x + 2y = 1$ which passes through the point $(-2, 1)$. Express your answer in slope-intercept form.

First, we need to find the slope of the given line $3x + 2y = 1$.

We can rewrite this equation in slope-intercept form:

$2y = -3x + 1$

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

The slope of this line is $m_1 = -\frac{3}{2}$.

The slope of a line perpendicular to this line is the negative reciprocal of $m_1$:

$m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{3}{2}} = \frac{2}{3}$

Now, we have the slope of the perpendicular line, $m_2 = \frac{2}{3}$, and a point it passes through, $(-2, 1)$.

We can use the point-slope form of a line:

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

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

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

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

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

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

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

3. Final Answer

(a) The y-intercept is $-0.5$.

(b) The slope of the line is $-0.7$.

(c) The equation of the line in standard form is $7x + 10y = -5$.

(4) The equation of the line perpendicular to $3x + 2y = 1$ and passing through $(-2, 1)$ is $y = \frac{2}{3}x + \frac{7}{3}$.