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Write the equation of the line that passes through [tex]$(-8, -4)$[/tex] and [tex]$(-6, -1)$[/tex] in slope-intercept form.

A. [tex]$y = -\frac{3}{2}x - 10$[/tex]

B. [tex]$y = \frac{2}{3}x + 3$[/tex]

C. [tex]$y = \frac{3}{2}x - 10$[/tex]

D. [tex]$y = \frac{3}{2}x + 8$[/tex]

Sagot :

GinnyAnsweridn
To find the equation of the line that passes through the points [tex]\((-8, -4)\)[/tex] and [tex]\((-6, -1)\)[/tex] in slope-intercept form ([tex]\(y = mx + b\)[/tex]), we need to follow these steps:

### Step 1: Find the Slope
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] can be found using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substitute the given points:
[tex]\[ (x_1, y_1) = (-8, -4) \][/tex]
[tex]\[ (x_2, y_2) = (-6, -1) \][/tex]

Now plug these values into the formula:
[tex]\[ m = \frac{-1 - (-4)}{-6 - (-8)} = \frac{-1 + 4}{-6 + 8} = \frac{3}{2} \][/tex]

So, the slope [tex]\(m = \frac{3}{2}\)[/tex].

### Step 2: Find the Y-Intercept
Using the slope-intercept form [tex]\(y = mx + b\)[/tex], we now need to find the y-intercept [tex]\(b\)[/tex].

We can use one of the given points to solve for [tex]\(b\)[/tex]. Let’s use the point [tex]\((-8, -4)\)[/tex]. Substitute [tex]\(x = -8\)[/tex], [tex]\(y = -4\)[/tex], and [tex]\(m = \frac{3}{2}\)[/tex] into the equation:
[tex]\[ -4 = \frac{3}{2}(-8) + b \][/tex]

Now solve for [tex]\(b\)[/tex]:
[tex]\[ -4 = \frac{3}{2} \times -8 + b \][/tex]
[tex]\[ -4 = -12 + b \][/tex]
[tex]\[ b = -4 + 12 \][/tex]
[tex]\[ b = 8 \][/tex]

So, the y-intercept [tex]\(b = 8\)[/tex].

### Step 3: Write the Equation
Now that we have both the slope [tex]\(m = \frac{3}{2}\)[/tex] and the y-intercept [tex]\(b = 8\)[/tex], we can write the equation of the line:
[tex]\[ y = \frac{3}{2}x + 8 \][/tex]

### Verification
Let’s quickly verify that this equation passes through both points:
1. For [tex]\((-8, -4)\)[/tex]:
[tex]\[ y = \frac{3}{2}(-8) + 8 = -12 + 8 = -4 \][/tex]
This matches the y-coordinate of the point.

2. For [tex]\((-6, -1)\)[/tex]:
[tex]\[ y = \frac{3}{2}(-6) + 8 = -9 + 8 = -1 \][/tex]
This matches the y-coordinate of the point.

Therefore, the correct equation of the line is:
[tex]\[ y = \frac{3}{2} x + 8 \][/tex]

The correct choice from the given options is:
[tex]\[ \boxed{y = \frac{3}{2} x + 8} \][/tex]
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