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Sagot :
To determine the equations of lines that are parallel to the given lines and pass through a specified point, we need to understand the characteristics of parallel lines:
Parallel lines have the same slope:
- For horizontal lines of the form [tex]\( y = c \)[/tex] (where [tex]\( c \)[/tex] is a constant), any line parallel to it will also be a horizontal line of the form [tex]\( y = k \)[/tex], where [tex]\( k \)[/tex] is another constant.
- For vertical lines of the form [tex]\( x = c \)[/tex] (where [tex]\( c \)[/tex] is a constant), any line parallel to it will also be a vertical line of the form [tex]\( x = k \)[/tex], where [tex]\( k \)[/tex] is another constant.
Given lines:
1. [tex]\( y = -2 \)[/tex]
2. [tex]\( x = -2 \)[/tex]
3. [tex]\( y = -4 \)[/tex]
4. [tex]\( x = -4 \)[/tex]
Now, let’s assume we need to find the equations of lines parallel to these given lines and passing through a specific given point [tex]\((1, 3)\)[/tex]:
1. Parallel to [tex]\( y = -2 \)[/tex]:
- The parallel line will be horizontal and have the form [tex]\( y = k \)[/tex].
- Since it must pass through the point [tex]\((1, 3)\)[/tex], the value of [tex]\( k \)[/tex] must be 3.
- Therefore, the equation of the line is [tex]\( y = 3 \)[/tex].
2. Parallel to [tex]\( x = -2 \)[/tex]:
- The parallel line will be vertical and have the form [tex]\( x = k \)[/tex].
- Since it must pass through the point [tex]\((1, 3)\)[/tex], the value of [tex]\( k \)[/tex] must be 1.
- Therefore, the equation of the line is [tex]\( x = 1 \)[/tex].
3. Parallel to [tex]\( y = -4 \)[/tex]:
- The parallel line will also be horizontal and have the form [tex]\( y = k \)[/tex].
- Since it must pass through the point [tex]\((1, 3)\)[/tex], the value of [tex]\( k \)[/tex] must be 3.
- Therefore, the equation of the line is [tex]\( y = 3 \)[/tex].
4. Parallel to [tex]\( x = -4 \)[/tex]:
- The parallel line will also be vertical and have the form [tex]\( x = k \)[/tex].
- Since it must pass through the point [tex]\((1, 3)\)[/tex], the value of [tex]\( k \)[/tex] must be 1.
- Therefore, the equation of the line is [tex]\( x = 1 \)[/tex].
So, the equations of the lines parallel to the given lines and passing through the point [tex]\((1, 3)\)[/tex] are:
- Parallel to [tex]\( y = -2 \)[/tex]: [tex]\( y = 3 \)[/tex]
- Parallel to [tex]\( x = -2 \)[/tex]: [tex]\( x = 1 \)[/tex]
- Parallel to [tex]\( y = -4 \)[/tex]: [tex]\( y = 3 \)[/tex]
- Parallel to [tex]\( x = -4 \)[/tex]: [tex]\( x = 1 \)[/tex]
Parallel lines have the same slope:
- For horizontal lines of the form [tex]\( y = c \)[/tex] (where [tex]\( c \)[/tex] is a constant), any line parallel to it will also be a horizontal line of the form [tex]\( y = k \)[/tex], where [tex]\( k \)[/tex] is another constant.
- For vertical lines of the form [tex]\( x = c \)[/tex] (where [tex]\( c \)[/tex] is a constant), any line parallel to it will also be a vertical line of the form [tex]\( x = k \)[/tex], where [tex]\( k \)[/tex] is another constant.
Given lines:
1. [tex]\( y = -2 \)[/tex]
2. [tex]\( x = -2 \)[/tex]
3. [tex]\( y = -4 \)[/tex]
4. [tex]\( x = -4 \)[/tex]
Now, let’s assume we need to find the equations of lines parallel to these given lines and passing through a specific given point [tex]\((1, 3)\)[/tex]:
1. Parallel to [tex]\( y = -2 \)[/tex]:
- The parallel line will be horizontal and have the form [tex]\( y = k \)[/tex].
- Since it must pass through the point [tex]\((1, 3)\)[/tex], the value of [tex]\( k \)[/tex] must be 3.
- Therefore, the equation of the line is [tex]\( y = 3 \)[/tex].
2. Parallel to [tex]\( x = -2 \)[/tex]:
- The parallel line will be vertical and have the form [tex]\( x = k \)[/tex].
- Since it must pass through the point [tex]\((1, 3)\)[/tex], the value of [tex]\( k \)[/tex] must be 1.
- Therefore, the equation of the line is [tex]\( x = 1 \)[/tex].
3. Parallel to [tex]\( y = -4 \)[/tex]:
- The parallel line will also be horizontal and have the form [tex]\( y = k \)[/tex].
- Since it must pass through the point [tex]\((1, 3)\)[/tex], the value of [tex]\( k \)[/tex] must be 3.
- Therefore, the equation of the line is [tex]\( y = 3 \)[/tex].
4. Parallel to [tex]\( x = -4 \)[/tex]:
- The parallel line will also be vertical and have the form [tex]\( x = k \)[/tex].
- Since it must pass through the point [tex]\((1, 3)\)[/tex], the value of [tex]\( k \)[/tex] must be 1.
- Therefore, the equation of the line is [tex]\( x = 1 \)[/tex].
So, the equations of the lines parallel to the given lines and passing through the point [tex]\((1, 3)\)[/tex] are:
- Parallel to [tex]\( y = -2 \)[/tex]: [tex]\( y = 3 \)[/tex]
- Parallel to [tex]\( x = -2 \)[/tex]: [tex]\( x = 1 \)[/tex]
- Parallel to [tex]\( y = -4 \)[/tex]: [tex]\( y = 3 \)[/tex]
- Parallel to [tex]\( x = -4 \)[/tex]: [tex]\( x = 1 \)[/tex]
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