Given that the line passes through the points (3, 4) and (5, 8), find the equation of the linear function.

To find the equation of a linear function given two points, we can use the point-slope form of a linear equation, which is:

\$"y" - "y"_1 = "m"("x" - "x"_1)\$

Where \$("x"_1, "y"_1)\$ are the coordinates of one of the points on the line, and m is the slope of the line.

The given points are

\$ ("x"_1, "y"_1) = (3, 4)\$
\$ ("x"_2, "y"_2) = (5,8) \$

Slope formula

\$ "m" = ("y"_2 - "y"_1)/("x"_2 - "x"_1) \$

By using the given points (3, 4) and (5, 8) then we get m value

⇒ \$ "m" = (8 - 4)/(5 - 3) = \cancel4^2/\cancel2 \$
⇒ m = 2

Now we can choose either of the two points to substitute into the equation. Let’s use (3, 4)

⇒ y - 4 = 2(x - 3)
⇒ y - 4 = 2x - 6

To get the equation in slope-intercept form (y = mx + b), we isolate y

⇒ y = 2x - 6 + 4
⇒ y = 2x - 2

Therefore, the equation of the linear function is y = 2x - 2.

This option represents the correct equation of the line passing through the given points, derived from the slope-intercept form with the correct slope and intercept

g(x) = 2x - 2

This option has an incorrect intercept; the slope is correct, but the y-intercept should be -2, not -3

g(x) = 2x - 3

This option also has an incorrect intercept; the slope is correct, but the y-intercept should be -2, not -4

g(x) = 2x - 4

This option has an incorrect intercept; the slope is correct, but the y-intercept should be -2, not -5

g(x) = 2x - 5

Option

A

Can you summarize what you’ve understood in the above steps?