Determine the equation of the linear relation in each of the following forms:

a) Point-slope form: [tex]y - y_1 = m(x - x_1)[/tex]

b) Slope-intercept form: [tex]y = mx + b[/tex]

c) General form: [tex]Ax + By + C = 0[/tex], where [tex]A, B, C \in \mathbb{I}[/tex], [tex]A \ \textgreater \ 0[/tex]



Answer :

To solve this problem, we need to determine the equation of a line that passes through a given point and has a specified slope in three different forms: point-slope form, slope-intercept form, and general form.

Given:
- Point: (1, 2)
- Slope: 3

### (a) Point-Slope Form

The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the given point and [tex]\( m \)[/tex] is the slope.

Plugging in the given point (1, 2) and the slope 3:
[tex]\[ y - 2 = 3(x - 1) \][/tex]

So, the point-slope form of the equation is:
[tex]\[ y - 2 = 3(x - 1) \][/tex]

### (b) Slope-Intercept Form

The slope-intercept form of the equation of a line is given by:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.

Using the point (1, 2) and the slope 3, we first need to determine [tex]\( b \)[/tex].

We know:
[tex]\[ y = 3x + b \][/tex]
At the point (1, 2):
[tex]\[ 2 = 3(1) + b \][/tex]
[tex]\[ 2 = 3 + b \][/tex]
[tex]\[ b = 2 - 3 \][/tex]
[tex]\[ b = -1 \][/tex]

So, the slope-intercept form of the equation is:
[tex]\[ y = 3x - 1 \][/tex]

### (c) General Form

The general form of the equation of a line is given by:
[tex]\[ Ax + By + C = 0 \][/tex]
where [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] are integers, and [tex]\( A > 0 \)[/tex].

Starting from the slope-intercept form [tex]\( y = 3x - 1 \)[/tex], we need to rearrange it into the general form.

[tex]\[ y = 3x - 1 \][/tex]
Subtract [tex]\( y \)[/tex] from both sides:
[tex]\[ 0 = 3x - y - 1 \][/tex]
Rewriting it for clarity:
[tex]\[ 3x - y - 1 = 0 \][/tex]

So, the general form of the equation is:
[tex]\[ 3x - y - 1 = 0 \][/tex]

In summary, the equations in the different forms are:

- Point-Slope Form: [tex]\( y - 2 = 3(x - 1) \)[/tex]
- Slope-Intercept Form: [tex]\( y = 3x - 1 \)[/tex]
- General Form: [tex]\( 3x - y - 1 = 0 \)[/tex]