Answer :
Sure! Let's derive the equation of the line that passes through the point (6, -3) and has a slope of [tex]\( \frac{1}{2} \)[/tex].
1. Identify the form of the equation:
The general form of a line's equation with slope [tex]\( m \)[/tex] and passing through point [tex]\((x_1, y_1) \)[/tex] is given by the point-slope form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( m = \frac{1}{2} \)[/tex], [tex]\( x_1 = 6 \)[/tex], and [tex]\( y_1 = -3 \)[/tex].
2. Substitute the values:
Plugging these values into the point-slope form, we get:
[tex]\[ y - (-3) = \frac{1}{2}(x - 6) \][/tex]
Simplifying this, we obtain:
[tex]\[ y + 3 = \frac{1}{2}(x - 6) \][/tex]
3. Convert to slope-intercept form:
To convert to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y + 3 = \frac{1}{2}x - \frac{6}{2} \][/tex]
[tex]\[ y + 3 = \frac{1}{2}x - 3 \][/tex]
4. Solve for [tex]\( y \)[/tex]:
Isolating [tex]\( y \)[/tex] on one side of the equation:
[tex]\[ y = \frac{1}{2}x - 3 - 3 \][/tex]
[tex]\[ y = \frac{1}{2}x - 6 \][/tex]
Hence, the equation of the line that passes through the point (6, -3) and has a slope of [tex]\(\frac{1}{2}\)[/tex] is:
[tex]\[ y = \frac{1}{2}x - 6 \][/tex]
So, the correct option is [tex]\(y = \frac{1}{2}x - 6\)[/tex].
1. Identify the form of the equation:
The general form of a line's equation with slope [tex]\( m \)[/tex] and passing through point [tex]\((x_1, y_1) \)[/tex] is given by the point-slope form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( m = \frac{1}{2} \)[/tex], [tex]\( x_1 = 6 \)[/tex], and [tex]\( y_1 = -3 \)[/tex].
2. Substitute the values:
Plugging these values into the point-slope form, we get:
[tex]\[ y - (-3) = \frac{1}{2}(x - 6) \][/tex]
Simplifying this, we obtain:
[tex]\[ y + 3 = \frac{1}{2}(x - 6) \][/tex]
3. Convert to slope-intercept form:
To convert to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y + 3 = \frac{1}{2}x - \frac{6}{2} \][/tex]
[tex]\[ y + 3 = \frac{1}{2}x - 3 \][/tex]
4. Solve for [tex]\( y \)[/tex]:
Isolating [tex]\( y \)[/tex] on one side of the equation:
[tex]\[ y = \frac{1}{2}x - 3 - 3 \][/tex]
[tex]\[ y = \frac{1}{2}x - 6 \][/tex]
Hence, the equation of the line that passes through the point (6, -3) and has a slope of [tex]\(\frac{1}{2}\)[/tex] is:
[tex]\[ y = \frac{1}{2}x - 6 \][/tex]
So, the correct option is [tex]\(y = \frac{1}{2}x - 6\)[/tex].
