Answer :
Sure, let's break down the solution step-by-step to find the equations for the line passing through the points [tex]\((-3, 5)\)[/tex] and [tex]\( (6, 2)\)[/tex].
### Step 1: Calculate the Slope (m)
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of our points:
[tex]\[ m = \frac{2 - 5}{6 - (-3)} = \frac{2 - 5}{6 + 3} = \frac{-3}{9} = -\frac{1}{3} \][/tex]
### Step 2: Write the Point-Slope Form of the Line
The point-slope form of the equation of the line is written as:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using one of our points [tex]\((-3, 5)\)[/tex] and the slope we just calculated:
[tex]\[ y - 5 = -\frac{1}{3}(x - (-3)) = -\frac{1}{3}(x + 3) \][/tex]
So, the point-slope form is:
[tex]\[ y - 5 = -\frac{1}{3}(x + 3) \][/tex]
### Step 3: Convert the Point-Slope Form to Slope-Intercept Form
The slope-intercept form of a linear equation is:
[tex]\[ y = mx + b \][/tex]
Let's start from the point-slope form and convert it step-by-step:
[tex]\[ y - 5 = -\frac{1}{3}(x + 3) \][/tex]
First, distribute the slope:
[tex]\[ y - 5 = -\frac{1}{3}x - 1 \][/tex]
Next, isolate [tex]\( y \)[/tex] by adding 5 to both sides of the equation:
[tex]\[ y = -\frac{1}{3}x - 1 + 5 \][/tex]
Simplify the constants on the right-hand side:
[tex]\[ y = -\frac{1}{3}x + 4 \][/tex]
Therefore, the slope-intercept form is:
[tex]\[ y = -\frac{1}{3}x + 4 \][/tex]
### Summary
- The point-slope form of the equation is:
[tex]\[ y - 5 = -\frac{1}{3}(x + 3) \][/tex]
- The slope-intercept form of the equation is:
[tex]\[ y = -\frac{1}{3}x + 4 \][/tex]
### Step 1: Calculate the Slope (m)
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of our points:
[tex]\[ m = \frac{2 - 5}{6 - (-3)} = \frac{2 - 5}{6 + 3} = \frac{-3}{9} = -\frac{1}{3} \][/tex]
### Step 2: Write the Point-Slope Form of the Line
The point-slope form of the equation of the line is written as:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using one of our points [tex]\((-3, 5)\)[/tex] and the slope we just calculated:
[tex]\[ y - 5 = -\frac{1}{3}(x - (-3)) = -\frac{1}{3}(x + 3) \][/tex]
So, the point-slope form is:
[tex]\[ y - 5 = -\frac{1}{3}(x + 3) \][/tex]
### Step 3: Convert the Point-Slope Form to Slope-Intercept Form
The slope-intercept form of a linear equation is:
[tex]\[ y = mx + b \][/tex]
Let's start from the point-slope form and convert it step-by-step:
[tex]\[ y - 5 = -\frac{1}{3}(x + 3) \][/tex]
First, distribute the slope:
[tex]\[ y - 5 = -\frac{1}{3}x - 1 \][/tex]
Next, isolate [tex]\( y \)[/tex] by adding 5 to both sides of the equation:
[tex]\[ y = -\frac{1}{3}x - 1 + 5 \][/tex]
Simplify the constants on the right-hand side:
[tex]\[ y = -\frac{1}{3}x + 4 \][/tex]
Therefore, the slope-intercept form is:
[tex]\[ y = -\frac{1}{3}x + 4 \][/tex]
### Summary
- The point-slope form of the equation is:
[tex]\[ y - 5 = -\frac{1}{3}(x + 3) \][/tex]
- The slope-intercept form of the equation is:
[tex]\[ y = -\frac{1}{3}x + 4 \][/tex]
