Answer :
To determine if the lines given by the equations [tex]\( y = 2x - 7 \)[/tex] and [tex]\( y = x - 7 \)[/tex] will intersect, we need to find a common point [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously.
### Step-by-Step Solution:
1. Set the Equations Equal:
Since both equations are equal to [tex]\( y \)[/tex], we can set them equal to each other to find the [tex]\( x \)[/tex]-coordinate of the intersection.
[tex]\[ 2x - 7 = x - 7 \][/tex]
2. Isolate [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], we subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ 2x - x - 7 = x - x - 7 \][/tex]
Simplifying this, we get:
[tex]\[ x - 7 = -7 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
Now, add 7 to both sides of the equation:
[tex]\[ x - 7 + 7 = -7 + 7 \][/tex]
Simplifying this, we get:
[tex]\[ x = 0 \][/tex]
4. Find the Corresponding [tex]\( y \)[/tex]-value:
Substitute [tex]\( x = 0 \)[/tex] back into either of the original equations to find the corresponding [tex]\( y \)[/tex]-value. Let's use [tex]\( y = 2x - 7 \)[/tex]:
[tex]\[ y = 2(0) - 7 \][/tex]
Simplifying this, we get:
[tex]\[ y = -7 \][/tex]
5. Intersection Point:
The intersection point of the two lines is [tex]\((0, -7)\)[/tex].
6. Verification:
To verify, we substitute [tex]\( x = 0 \)[/tex] into both original equations to ensure the point lies on both lines.
For the first equation, [tex]\( y = 2x - 7 \)[/tex]:
[tex]\[ y = 2(0) - 7 = -7 \][/tex]
For the second equation, [tex]\( y = x - 7 \)[/tex]:
[tex]\[ y = 0 - 7 = -7 \][/tex]
Since both equations give the same [tex]\( y \)[/tex]-value of [tex]\(-7\)[/tex], the lines indeed intersect at the point [tex]\((0, -7)\)[/tex].
### Conclusion:
Yes, the lines given by the equations [tex]\( y = 2x - 7 \)[/tex] and [tex]\( y = x - 7 \)[/tex] will intersect. They intersect at the point [tex]\((0, -7)\)[/tex]. Thus, the lines do intersect.
### Step-by-Step Solution:
1. Set the Equations Equal:
Since both equations are equal to [tex]\( y \)[/tex], we can set them equal to each other to find the [tex]\( x \)[/tex]-coordinate of the intersection.
[tex]\[ 2x - 7 = x - 7 \][/tex]
2. Isolate [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], we subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ 2x - x - 7 = x - x - 7 \][/tex]
Simplifying this, we get:
[tex]\[ x - 7 = -7 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
Now, add 7 to both sides of the equation:
[tex]\[ x - 7 + 7 = -7 + 7 \][/tex]
Simplifying this, we get:
[tex]\[ x = 0 \][/tex]
4. Find the Corresponding [tex]\( y \)[/tex]-value:
Substitute [tex]\( x = 0 \)[/tex] back into either of the original equations to find the corresponding [tex]\( y \)[/tex]-value. Let's use [tex]\( y = 2x - 7 \)[/tex]:
[tex]\[ y = 2(0) - 7 \][/tex]
Simplifying this, we get:
[tex]\[ y = -7 \][/tex]
5. Intersection Point:
The intersection point of the two lines is [tex]\((0, -7)\)[/tex].
6. Verification:
To verify, we substitute [tex]\( x = 0 \)[/tex] into both original equations to ensure the point lies on both lines.
For the first equation, [tex]\( y = 2x - 7 \)[/tex]:
[tex]\[ y = 2(0) - 7 = -7 \][/tex]
For the second equation, [tex]\( y = x - 7 \)[/tex]:
[tex]\[ y = 0 - 7 = -7 \][/tex]
Since both equations give the same [tex]\( y \)[/tex]-value of [tex]\(-7\)[/tex], the lines indeed intersect at the point [tex]\((0, -7)\)[/tex].
### Conclusion:
Yes, the lines given by the equations [tex]\( y = 2x - 7 \)[/tex] and [tex]\( y = x - 7 \)[/tex] will intersect. They intersect at the point [tex]\((0, -7)\)[/tex]. Thus, the lines do intersect.
