Question

33. For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.

33. Passes through (1, 5) and (4, 11)

Answer 1

The linear equation for the line which passes through the points given as (1,5) and (4,11), is written in the point-slope form as y=2x+3.

Explanation:

A condition is given that a line passes through the points whose coordinates are (1,5) and (4,11).

It is asked to find the linear equation which satisfies the given condition.

Step 1 of 2

Determine the slope of the line.

The points through which the line passes are given as (1,5) and (4,11). Next, the formula for the slope is given as,

`$m=(y_(2)-y_(1))/(x_(2)-x_(1))$`

Substitute 11&5 for
`$y_(2)$` and
`$y_(1)$` respectively, and
`$4 \& 1$` for
`$x_(2)$` and
`$x_(1)$` respectively in the above formula. Then simplify to get the slope as follows,

`$$\begin{aligned}m &=(11-5)/(4-1) \\m &=(6)/(3) \\m &=2\end{aligned}$$`

Step 2 of 2

Write the linear equation in point-slope form.

A linear equation in point slope form is given as,

`$$y-y_(1)=m\left(x-x_(1)\right)$$`

Substitute 2 for
`$m, 1$` for
`$x_(1)$`, and 5 for
`$y_(1)$` in the above equation and simplify using the distributive property as follows,

`$$\begin{aligned}&y-5=2(x-1) \\&y-5=2 x-2 \\&y=2 x-2+5 \\&y=2 x+3\end{aligned}$$`

This is the required linear equation.