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Write an equation in point-slope form of the line that passes through the given points, then write the equation in slope-intercept form.

Points: (-3, 5) and (6, 2)

1. What is the point-slope form of the equation of the line?

[tex]\( y - 5 = -\frac{1}{3}(x + 3) \)[/tex]

(Simplify your answer. Use integers or fractions for any numbers in the equation.)

2. What is the slope-intercept form of the equation of the line?

[tex]\(\boxed{\text{(Simplify your answer. Use integers or fractions for any numbers in the equation.)}}\)[/tex]

Sagot :

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]
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