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Choose the equation below that represents the line passing through the point [tex][tex]$(-5, 1)$[/tex][/tex] with a slope of [tex]\frac{3}{2}[/tex].

A. [tex]y - 5 = \frac{3}{2}(x + 1)[/tex]

B. [tex]y + 1 = \frac{3}{2}(x - 5)[/tex]

C. [tex]y + 5 = \frac{3}{2}(x - 1)[/tex]

D. [tex]y - 1 = \frac{3}{2}(x + 5)[/tex]


Sagot :

To find the equation of the line passing through the point [tex]\((-5, 1)\)[/tex] with a slope of [tex]\(\frac{3}{2}\)[/tex], we can use the point-slope form of the equation of a line. The point-slope form is given by:

[tex]\[ y - y_1 = m(x - x_1) \][/tex]

where [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope of the line. In this problem, the point [tex]\((-5, 1)\)[/tex] is given along with the slope [tex]\(m = \frac{3}{2}\)[/tex].

Plugging in the given point and slope into the point-slope form equation, we have:

[tex]\[ y - 1 = \frac{3}{2}(x - (-5)) \][/tex]

Since subtracting a negative number is equivalent to adding the positive counterpart, we can simplify the equation:

[tex]\[ y - 1 = \frac{3}{2}(x + 5) \][/tex]

Therefore, the correct equation of the line is:

[tex]\[ y - 1 = \frac{3}{2}(x + 5) \][/tex]

Comparing this with the options provided:

1. [tex]\(y - 5 = \frac{3}{2}(x + 1)\)[/tex]
2. [tex]\(y + 1 = \frac{3}{2}(x - 5)\)[/tex]
3. [tex]\(y + 5 = \frac{3}{2}(x - 1)\)[/tex]
4. [tex]\(y - 1 = \frac{3}{2}(x + 5)\)[/tex]

We see that option 4 matches our derived equation. Hence, the correct choice is:

[tex]\[ \boxed{y - 1 = \frac{3}{2}(x + 5)} \][/tex]