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Write an equation of the line that passes through the points [tex](-3,1)[/tex] and [tex](-2,-5)[/tex]:

A. [tex]y = 6x - 17[/tex]
B. [tex]y = -6x - 17[/tex]
C. [tex]y = -6x + 17[/tex]
D. [tex]y = -6x + \frac{1}{17}[/tex]


Sagot :

Sure, let's determine the equation of the line that passes through the points [tex]\((-3, 1)\)[/tex] and [tex]\((-2, -5)\)[/tex].

### Step 1: Find the slope (m) of the line.

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 calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substitute the points [tex]\((-3, 1)\)[/tex] and [tex]\((-2, -5)\)[/tex] into the formula:
[tex]\[ m = \frac{-5 - 1}{-2 - (-3)} = \frac{-5 - 1}{-2 + 3} = \frac{-6}{1} = -6 \][/tex]

So, the slope [tex]\(m\)[/tex] is [tex]\(-6\)[/tex].

### Step 2: Find the y-intercept (b) of the line.

The y-intercept [tex]\(b\)[/tex] can be found using the point-slope form of the line equation [tex]\(y = mx + b\)[/tex]. We can use one of the points and the calculated slope to find [tex]\(b\)[/tex].

Let's use the point [tex]\((-3, 1)\)[/tex]. Substitute [tex]\(x = -3\)[/tex], [tex]\(y = 1\)[/tex], and [tex]\(m = -6\)[/tex] into the equation [tex]\(y = mx + b\)[/tex]:
[tex]\[ 1 = -6(-3) + b \][/tex]
Simplify:
[tex]\[ 1 = 18 + b \][/tex]
Subtract 18 from both sides to solve for [tex]\(b\)[/tex]:
[tex]\[ b = 1 - 18 \][/tex]
[tex]\[ b = -17 \][/tex]

So, the y-intercept [tex]\(b\)[/tex] is [tex]\(-17\)[/tex].

### Step 3: Write the equation of the line.

Now that we have the slope [tex]\(m = -6\)[/tex] and the y-intercept [tex]\(b = -17\)[/tex], we can write the equation of the line:
[tex]\[ y = -6x - 17 \][/tex]

### Step 4: Compare with the given options.

Let's match this equation with the provided options:

a. [tex]\(y=6x-17\)[/tex]
b. [tex]\(y=-6x-17\)[/tex]
c. [tex]\(y=-6x+17\)[/tex]
d. [tex]\(y=-6x+\frac{1}{17}\)[/tex]

The correct equation is:
[tex]\[ y = -6x - 17 \][/tex]

Therefore, the answer is:
b. [tex]\(y = -6x - 17\)[/tex]