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Write the equation of the line that passes through the points [tex]\((3, 6)\)[/tex] and [tex]\((4, 10)\)[/tex] using function notation.

A. [tex]\(f(x) = 4x - 6\)[/tex]
B. [tex]\(f(x) = x + 4\)[/tex]
C. [tex]\(y = x + 4\)[/tex]
D. [tex]\(y = 4x - 6\)[/tex]


Sagot :

To determine the equation of the line that passes through the points [tex]\((3,6)\)[/tex] and [tex]\((4,10)\)[/tex], we need to follow these steps:

1. Calculate the slope ([tex]\(m\)[/tex]) of the line:

The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substituting the given points [tex]\((3, 6)\)[/tex] and [tex]\((4, 10)\)[/tex] into the formula:
[tex]\[ m = \frac{10 - 6}{4 - 3} = \frac{4}{1} = 4 \][/tex]

2. Find the y-intercept ([tex]\(b\)[/tex]) using the slope-intercept form of the equation:

The slope-intercept form of the equation of a line is:
[tex]\[ y = mx + b \][/tex]

To find [tex]\(b\)[/tex], we can use one of the given points. Here, we use the point [tex]\((3, 6)\)[/tex]. Plug in the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(m\)[/tex]:
[tex]\[ 6 = 4 \cdot 3 + b \][/tex]

Solving for [tex]\(b\)[/tex]:
[tex]\[ 6 = 12 + b \][/tex]
[tex]\[ b = 6 - 12 \][/tex]
[tex]\[ b = -6 \][/tex]

3. Write the equation in function notation [tex]\(f(x)\)[/tex]:

Therefore, substituting the values of [tex]\(m\)[/tex] and [tex]\(b\)[/tex] into the equation [tex]\(y = mx + b\)[/tex], we get:
[tex]\[ y = 4x - 6 \][/tex]

In function notation, we write this as:
[tex]\[ f(x) = 4x - 6 \][/tex]

Thus, the equation of the line that passes through the points [tex]\((3, 6)\)[/tex] and [tex]\((4, 10)\)[/tex] is:
[tex]\[ f(x) = 4x - 6 \][/tex]

Among the given options, the correct one is:
[tex]\( y = 4x - 6 \)[/tex].
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