Certainly! Let's address each part of the question step-by-step.
### Part (a):
You need to find the equation of a straight line with a given gradient (slope) and y-intercept.
1. Gradient (Slope): The gradient is given as [tex]\(3\)[/tex].
2. Y-intercept: The y-intercept is given as [tex]\(-4\)[/tex].
The general form of the equation of a straight line is:
[tex]\[ y = mx + c \][/tex]
where:
- [tex]\(m\)[/tex] is the gradient
- [tex]\(c\)[/tex] is the y-intercept
Plugging in the given values, we get:
[tex]\[ y = 3x + (-4) \][/tex]
So, the equation of the line for part (a) is:
[tex]\[ y = 3x - 4 \][/tex]
### Part (b):
You need to find the equation of a straight line with a given angle of inclination and y-intercept.
1. Angle of Inclination: The angle of inclination is given as [tex]\(135^\circ\)[/tex].
2. Y-intercept: The y-intercept is given as [tex]\(5\)[/tex].
The gradient of a line can be found using the tangent of the angle of inclination:
[tex]\[ m = \tan(\theta) \][/tex]
where:
- [tex]\(\theta\)[/tex] is the angle of inclination.
For [tex]\(\theta = 135^\circ\)[/tex]:
[tex]\[ m = \tan(135^\circ) \][/tex]
The value of [tex]\(\tan(135^\circ)\)[/tex] is [tex]\(-1.0000000000000002\)[/tex] (which is approximately [tex]\(-1\)[/tex]).
The general form of the equation of a straight line is:
[tex]\[ y = mx + c \][/tex]
Plugging in the calculated gradient and the given y-intercept, we get:
[tex]\[ y = -1.0000000000000002x + 5 \][/tex]
So, the equation of the line for part (b) is:
[tex]\[ y = -1.0000000000000002x + 5 \][/tex]
To summarize:
- The equation of the line with a gradient of 3 and a y-intercept of -4 is:
[tex]\[ y = 3x - 4 \][/tex]
- The equation of the line with an angle of inclination of 135° and a y-intercept of 5 is:
[tex]\[ y = -1.0000000000000002x + 5 \][/tex]
