The problem has two parts. Part 1: Find the equation of the lines in the form $y = mx + c$ for the given graphs. Part 2: Find the x and y-intercepts and sketch the graph for the given equations: $y = x + 4$ and $y = -2x - 6$.

1. Problem Description

The problem has two parts.

Part 1: Find the equation of the lines in the form $y = mx + c$ for the given graphs.

Part 2: Find the x and y-intercepts and sketch the graph for the given equations: $y = x + 4$ and $y = -2x - 6$.

2. Solution Steps

Part 1a:

The graph passes through the points $(0, 3)$ and $(3, 0)$.

The y-intercept is $c = 3$.

The slope $m$ can be calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{3 - 0} = \frac{-3}{3} = -1$

So, the equation of the line is $y = -x + 3$.

Part 1b:

The graph passes through the points $(0, 1)$ and $(1, 4)$.

The y-intercept is $c = 1$.

The slope $m$ can be calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{1 - 0} = \frac{3}{1} = 3$

So, the equation of the line is $y = 3x + 1$.

Part 2a:

Given equation: $y = x + 4$

To find the y-intercept, set $x = 0$:

$y = 0 + 4 = 4$.

So, the y-intercept is $(0, 4)$.

To find the x-intercept, set $y = 0$:

$0 = x + 4$

$x = -4$.

So, the x-intercept is $(-4, 0)$.

Part 2b:

Given equation: $y = -2x - 6$

To find the y-intercept, set $x = 0$:

$y = -2(0) - 6 = -6$.

So, the y-intercept is $(0, -6)$.

To find the x-intercept, set $y = 0$:

$0 = -2x - 6$

$2x = -6$

$x = -3$.

So, the x-intercept is $(-3, 0)$.

3. Final Answer

Part 1a: $y = -x + 3$

Part 1b: $y = 3x + 1$

Part 2a: x-intercept: $(-4, 0)$, y-intercept: $(0, 4)$

Part 2b: x-intercept: $(-3, 0)$, y-intercept: $(0, -6)$