To find the equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept, we can follow these steps.
We know:
- A point on the line: [tex]\( (2, -4) \)[/tex]
- The slope of the line: [tex]\( \frac{5}{2} \)[/tex]
Step 1: Identify the point-slope form of the line:
The point-slope form of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the point on the line and [tex]\( m \)[/tex] is the slope.
Substitute [tex]\( (x_1, y_1) = (2, -4) \)[/tex] and [tex]\( m = \frac{5}{2} \)[/tex]:
[tex]\[ y - (-4) = \frac{5}{2}(x - 2) \][/tex]
Step 2: Simplify the equation:
First, simplify [tex]\( y - (-4) \)[/tex] to [tex]\( y + 4 \)[/tex]:
[tex]\[ y + 4 = \frac{5}{2}(x - 2) \][/tex]
Step 3: Distribute the slope on the right hand side of the equation:
[tex]\[ y + 4 = \frac{5}{2}x - \frac{5}{2} \cdot 2 \][/tex]
[tex]\[ y + 4 = \frac{5}{2}x - 5 \][/tex]
Step 4: Solve for [tex]\( y \)[/tex]:
To get [tex]\( y \)[/tex] by itself, subtract 4 from both sides:
[tex]\[ y = \frac{5}{2}x - 5 - 4 \][/tex]
[tex]\[ y = \frac{5}{2}x - 9 \][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[ y = \frac{5}{2}x - 9 \][/tex]
