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What is the point-slope form of a line that has a slope of 3 and passes through the point [tex]$(-1, 4)$[/tex]?

A. [tex]$y - (-1) = 5(x - 4)$[/tex]

B. [tex][tex]$4 - y_1 = 3(-1 - x_1)$[/tex][/tex]

C. [tex]$y - 3 = -1(x - (-4))$[/tex]

D. [tex]$y - 4 = 3(x - (-1))$[/tex]

Sagot :

GinnyAnsweridn
To find the point-slope form of a line with a given slope and a point that the line passes through, we use the point-slope formula of a linear equation:

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

Where:
- [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
- [tex]\( m \)[/tex] is the slope of the line.

Given the slope [tex]\( m = 3 \)[/tex] and the point [tex]\((-1, 4)\)[/tex], we substitute these values into the point-slope formula.

First, identify the values for the slope [tex]\( m \)[/tex] and the coordinates [tex]\((x_1, y_1)\)[/tex]:
- Slope, [tex]\( m = 3 \)[/tex]
- Point, [tex]\( (x_1, y_1) = (-1, 4) \)[/tex]

Now, substitute these values into the formula:

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

Substitute [tex]\( y_1 = 4 \)[/tex], [tex]\( m = 3 \)[/tex], and [tex]\( x_1 = -1 \)[/tex]:

[tex]\[ y - 4 = 3(x - (-1)) \][/tex]

This simplifies to:

[tex]\[ y - 4 = 3(x + 1) \][/tex]

Thus, the point-slope form of the line that has a slope of 3 and passes through the point [tex]\((-1, 4)\)[/tex] is:

[tex]\[ y - 4 = 3(x + 1) \][/tex]

Among the given options, this matches:

[tex]\[ y - 4 = 3[(x - (-1))] \][/tex]

So, the correct answer is:
[tex]\[ y - 4 = 3[(x - (-1))] \][/tex]
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