To find the equation of a line given the gradient (slope) \( m \) and the \( y \)-intercept \( b \), we use the slope-intercept form of a linear equation:
[tex]\[ y = mx + b \][/tex]
Let's determine the equation for each line step-by-step:
### (a) Gradient \( m = 2 \), \( y \)-intercept \( b = 4 \)
Using the slope-intercept form:
[tex]\[ y = 2x + 4 \][/tex]
So, the equation is:
[tex]\[ y = 2x + 4 \][/tex]
### (b) Gradient \( m = -2 \), \( y \)-intercept \( b = -4 \)
Using the slope-intercept form:
[tex]\[ y = -2x - 4 \][/tex]
So, the equation is:
[tex]\[ y = -2x - 4 \][/tex]
### (c) Gradient \( m = 1 \), \( y \)-intercept \( b = -\frac{1}{5} \)
Using the slope-intercept form:
[tex]\[ y = 1x - \frac{1}{5} \][/tex]
So, the equation is:
[tex]\[ y = x - \frac{1}{5} \][/tex]
### (d) Gradient \( m = -1 \), \( y \)-intercept \( b = 3.78 \)
Using the slope-intercept form:
[tex]\[ y = -1x + 3.78 \][/tex]
So, the equation is:
[tex]\[ y = -x + 3.78 \][/tex]
### (e) Gradient \( m = -\frac{2}{3} \), \( y \)-intercept \( b = 0 \)
Using the slope-intercept form:
[tex]\[ y = -\frac{2}{3}x + 0 \][/tex]
Since the \( y \)-intercept is zero, we can simplify this to:
[tex]\[ y = -\frac{2}{3}x \][/tex]
So, the equation is:
[tex]\[ y = -\frac{2}{3}x \][/tex]
### (f) Gradient \( m = 0 \), \( y \)-intercept \( b = -\frac{2}{3} \)
Using the slope-intercept form:
[tex]\[ y = 0x - \frac{2}{3} \][/tex]
Since the gradient is zero, the line is horizontal and the equation simplifies to:
[tex]\[ y = -\frac{2}{3} \][/tex]
So, the equation is:
[tex]\[ y = -\frac{2}{3} \][/tex]
To summarize, the equations of the lines are:
(a) \( y = 2x + 4 \)
(b) \( y = -2x - 4 \)
(c) \( y = x - \frac{1}{5} \)
(d) \( y = -x + 3.78 \)
(e) \( y = -\frac{2}{3}x \)
(f) [tex]\( y = -\frac{2}{3} \)[/tex]
