To determine the relation between the two lines, we first need to see if both equations are in slope-intercept form, which is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope and [tex]\( b \)[/tex] represents the y-intercept.
The first equation is already in the slope-intercept form:
[tex]\[ y = 4x - 5 \][/tex]
This equation has a slope of [tex]\( m = 4 \)[/tex] and a y-intercept of [tex]\( b = -5 \)[/tex].
The second equation is:
[tex]\[ 8x + y = 3 \][/tex]
To convert it into slope-intercept form, we isolate [tex]\( y \)[/tex]:
[tex]\[ y = -8x + 3 \][/tex]
Now it's in the slope-intercept form with a slope [tex]\( m = -8 \)[/tex] and y-intercept [tex]\( b = 3 \)[/tex].
The slopes of the two lines are:
- Slope of the first line: [tex]\( m = 4 \)[/tex]
- Slope of the second line: [tex]\( m = -8 \)[/tex]
These slopes are not equal to each other. Since the slopes are different, the lines are neither parallel nor the same line; they are intersecting. This means:
- The two lines are intersecting, which answers part (a).
Now let's consider part (b) regarding the number of solutions:
Because the lines are intersecting, they meet at exactly one point. Therefore, the system of equations will have exactly one solution. Hence for part (b), the answer is:
- The system of equations will have one solution.
