Question

1. identify which form each equation is given then find the requested information.

forms: slope - intercept, standard, or point - slope

a) $3x - 6y = 18$ \t\tform: \t\tfind the x and y intercepts

b) $y + 8 = - 3(x - 5)$ \tform: \t\tidentify the slope and point

c) $y = -\frac{3}{5}x + 5$ \t\tform: \t\tidentify the slope and y - intercept

Explanation:

Part (a)

Step 1: Identify the form

The equation \(3x - 6y = 18\) is in standard form because the standard form of a linear equation is \(Ax + By = C\) (where \(A\), \(B\), and \(C\) are integers, and \(A\geq0\)). Here, \(A = 3\), \(B=- 6\), \(C = 18\).

Step 2: Find the x - intercept

To find the x - intercept, set \(y = 0\) in the equation \(3x-6y = 18\).
Substitute \(y = 0\) into the equation:
\(3x-6(0)=18\)
\(3x=18\)
Divide both sides by 3: \(x=\frac{18}{3}=6\)
So the x - intercept is \((6,0)\).

Step 3: Find the y - intercept

To find the y - intercept, set \(x = 0\) in the equation \(3x-6y = 18\).
Substitute \(x = 0\) into the equation:
\(3(0)-6y=18\)
\(-6y = 18\)
Divide both sides by \(- 6\): \(y=\frac{18}{-6}=-3\)
So the y - intercept is \((0, - 3)\).

Step 1: Identify the form

The equation \(y + 8=-3(x - 5)\) is in point - slope form. The point - slope form of a linear equation is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope. In the equation \(y+8=-3(x - 5)\), we can rewrite it as \(y-(-8)=-3(x - 5)\), so it is in point - slope form.

Step 2: Identify the slope and the point

Comparing \(y-(-8)=-3(x - 5)\) with \(y - y_1=m(x - x_1)\), we have:
The slope \(m=-3\) and the point \((x_1,y_1)=(5,-8)\)

Step 1: Identify the form

The equation \(y=-\frac{3}{5}x + 5\) is in slope - intercept form. The slope - intercept form of a linear equation is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. Here, \(m =-\frac{3}{5}\) and \(b = 5\).

Step 2: Identify the slope and the y - intercept

From the equation \(y=-\frac{3}{5}x + 5\) (in the form \(y=mx + b\)):
The slope \(m=-\frac{3}{5}\) and the y - intercept \(b = 5\) (the y - intercept is the point \((0,5)\))

Answer:

(a):
Form: Standard form; x - intercept: \((6,0)\); y - intercept: \((0,-3)\)

Part (b)