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State the equation of each of the following lines given its gradient and [tex]$y$[/tex]-intercept.

(a) Gradient \(= 2\), \(y\)-intercept \(= 4\)

(b) Gradient \(= -2\), \(y\)-intercept \(= -4\)

(c) Gradient \(= 1\), \(y\)-intercept \(= -\frac{1}{5}\)

(d) Gradient \(= -1\), \(y\)-intercept \(= 3.78\)

(e) Gradient \(= -\frac{2}{3}\), \(y\)-intercept \(= 0\)

(f) Gradient [tex]\(= 0\)[/tex], [tex]\(y\)[/tex]-intercept [tex]\(= -\frac{2}{3}\)[/tex]

Sagot :

GinnyAnswer
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]
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