Sure, let's work through this problem step-by-step to find the slope-intercept form of the equation of the line.
### Step 1: Understand the problem
We are given:
- A point through which the line passes: [tex]\((3, -5)\)[/tex]
- The slope of the line: [tex]\( -\frac{2}{3} \)[/tex]
### Step 2: Recall the slope-intercept form
The slope-intercept form of the equation of a line is:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope
- [tex]\( b \)[/tex] is the y-intercept
### Step 3: Substitute the slope and the point into the equation
We need to find the y-intercept [tex]\((b)\)[/tex]. To do this, we use the given point [tex]\((x_1, y_1) = (3, -5)\)[/tex] and the slope [tex]\( m = -\frac{2}{3} \)[/tex].
Substitute the given point and the slope into the slope-intercept form:
[tex]\[ y_1 = mx_1 + b \][/tex]
[tex]\[ -5 = -\frac{2}{3} \cdot 3 + b \][/tex]
### Step 4: Solve for [tex]\( b \)[/tex]
First, simplify the multiplication:
[tex]\[ -5 = -2 + b \][/tex]
Then, solve for [tex]\( b \)[/tex]:
[tex]\[ -5 + 2 = b \][/tex]
[tex]\[ b = -3 \][/tex]
### Step 5: Write the final equation
Now that we have the y-intercept [tex]\( b = -3 \)[/tex] and the slope [tex]\( m = -\frac{2}{3} \)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[ y = -\frac{2}{3} x - 3 \][/tex]
### Final Answer
The equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{2}{3}x - 3 \][/tex]
