To write the equation of a line in slope-intercept form, which is [tex]\( y = mx + b \)[/tex], we need two pieces of information: the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex].
Given:
- The slope [tex]\( m = -\frac{7}{12} \)[/tex].
- A point the line passes through: [tex]\((-6, 12)\)[/tex].
We start by substituting the slope [tex]\( m \)[/tex] and the coordinates of the given point [tex]\((x, y) = (-6, 12)\)[/tex] into the slope-intercept form equation and solve for [tex]\( b \)[/tex] (the y-intercept).
The slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Substitute [tex]\( x = -6 \)[/tex], [tex]\( y = 12 \)[/tex], and [tex]\( m = -\frac{7}{12} \)[/tex] into the equation:
[tex]\[ 12 = \left(-\frac{7}{12}\right)(-6) + b \][/tex]
Next, we calculate [tex]\( -\frac{7}{12} \times -6 \)[/tex]:
[tex]\[ -\frac{7}{12} \times -6 = \frac{7}{2} \][/tex]
So, the equation becomes:
[tex]\[ 12 = \frac{7}{2} + b \][/tex]
To find [tex]\( b \)[/tex], isolate [tex]\( b \)[/tex] by subtracting [tex]\(\frac{7}{2}\)[/tex] from both sides:
[tex]\[ 12 - \frac{7}{2} = b \][/tex]
Convert [tex]\( 12 \)[/tex] to a fraction with the same denominator as [tex]\(\frac{7}{2}\)[/tex]:
[tex]\[ 12 = \frac{24}{2} \][/tex]
Subtract the fractions:
[tex]\[ \frac{24}{2} - \frac{7}{2} = \frac{17}{2} \][/tex]
Thus,
[tex]\[ b = \frac{17}{2} \][/tex]
Which simplifies to:
[tex]\[ b = 8.5 \][/tex]
Now, we have the slope [tex]\( m = -\frac{7}{12} \)[/tex] and the y-intercept [tex]\( b = 8.5 \)[/tex].
Therefore, the equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{7}{12}x + 8.5 \][/tex]
