Answer :
To identify the equation of the line that is parallel to the line [tex]\( y = -3x + 2 \)[/tex] and passes through the point [tex]\((-4, -5)\)[/tex], follow these steps:
1. Determine the slope of the given line:
The equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
For the line [tex]\( y = -3x + 2 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\(-3\)[/tex].
2. Understand the properties of parallel lines:
Parallel lines have the same slope. So, the line parallel to [tex]\( y = -3x + 2 \)[/tex] will also have a slope of [tex]\(-3\)[/tex].
3. Use the point-slope form of the equation of a line:
The point-slope form is given by [tex]\( y - y_1 = m(x - x_1) \)[/tex],
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
Here, [tex]\((x_1, y_1) = (-4, -5)\)[/tex] and [tex]\( m = -3 \)[/tex].
4. Substitute the slope and the given point into the point-slope form:
[tex]\[ y - (-5) = -3 (x - (-4)) \][/tex]
Simplify the equation:
[tex]\[ y + 5 = -3 (x + 4) \][/tex]
5. Distribute the slope ( [tex]\(-3\)[/tex] ):
[tex]\[ y + 5 = -3x - 12 \][/tex]
6. Isolate [tex]\( y \)[/tex] to put the equation in the slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[ y = -3x - 12 - 5 \][/tex]
[tex]\[ y = -3x - 17 \][/tex]
Therefore, the equation of the line that is parallel to [tex]\( y = -3x + 2 \)[/tex] and passing through [tex]\((-4, -5)\)[/tex] is [tex]\( y = -3x - 17 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{y = -3x - 17} \][/tex]
1. Determine the slope of the given line:
The equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
For the line [tex]\( y = -3x + 2 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\(-3\)[/tex].
2. Understand the properties of parallel lines:
Parallel lines have the same slope. So, the line parallel to [tex]\( y = -3x + 2 \)[/tex] will also have a slope of [tex]\(-3\)[/tex].
3. Use the point-slope form of the equation of a line:
The point-slope form is given by [tex]\( y - y_1 = m(x - x_1) \)[/tex],
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
Here, [tex]\((x_1, y_1) = (-4, -5)\)[/tex] and [tex]\( m = -3 \)[/tex].
4. Substitute the slope and the given point into the point-slope form:
[tex]\[ y - (-5) = -3 (x - (-4)) \][/tex]
Simplify the equation:
[tex]\[ y + 5 = -3 (x + 4) \][/tex]
5. Distribute the slope ( [tex]\(-3\)[/tex] ):
[tex]\[ y + 5 = -3x - 12 \][/tex]
6. Isolate [tex]\( y \)[/tex] to put the equation in the slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[ y = -3x - 12 - 5 \][/tex]
[tex]\[ y = -3x - 17 \][/tex]
Therefore, the equation of the line that is parallel to [tex]\( y = -3x + 2 \)[/tex] and passing through [tex]\((-4, -5)\)[/tex] is [tex]\( y = -3x - 17 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{y = -3x - 17} \][/tex]
