# Need some help to write the equations of these lines

• Tak.Phy
In summary, the conversation is about finding the equations of straight lines using two points on the line. A straight line can be written as y = ax + b, where a and b can be found by substituting the coordinates of two points on the line into the equation. The process is demonstrated using an example of a line going through (1,3) and (3,0), resulting in the equation y = -(3/2)x + 9/2 or 2y = -3x + 9 or 3x + 2y = 9. It is also mentioned that for a vertical line, the equation is written as x = constant. The conversation ends with a clarification on finding the slope and the value
Tak.Phy
Hi everyone
So I have this homework and I need some help.
I did not know how to write the equations here in the topic so I thought the best solution is to upload a picture of the notebook page.

These are all straight lines. Don't you know the equation of a straight line?

Geometrically, a straight line is determined by two points. Any (non-vertical) line can be written y= ax+ b. Choose two points on each line. Replacing x and y with the x and y coordinates of those points gives you two equations to solve for a and b.

For example, if one line goes through (1, 3) and (3, 0) then 3= a(1)+ b and 0= a(3)+ b so we have the equations a+ b= 3 and 3a+ b= 0. Subtract the first equation from the second to get 2a= -3 so that a= -3/2. Then the first equation becomes a+ b= (-3/2)+ b= 3 so b= 3+ 3/2= 9/2. With a=-3/2 and b= 9/2, the equation is y= -(3/2)x+ 9/2 which could also be written 2y= -3x+ 9 or 3x+2y= 9.

(A vertical line can be written "x= constant".)

Thanks Hallsoflvy.

P.S: My cousin used my laptop and it seems that he posted this topic. I am very very sorry about this and it will never happen again.

Imho that's a little confusing. to find slope remember its m=$\frac{y_2 - y_1}{x_2-x_1}$ where it doesn't matter which order you use for y2 or y1. So to use example of line with (1,3) and (3,0), you could do it:

m = $\frac{0-3}{3-1}$ ⇔ $\frac{3-0}{1-3}$ = -$\frac{3}{2}$

y = mx+b, we want to find b, we already have m and can use one of the points above for (x,y):
0 = (-$\frac{3}{2}$)(3)+b → b = $\frac{9}{2}$

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## What are the equations of these lines?

The equations of lines can be written in the slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept. Another form is the point-slope form, y - y1 = m(x - x1), where (x1, y1) is a point on the line.

## How do I determine the slope of a line?

The slope of a line can be calculated by taking the change in y-coordinates divided by the change in x-coordinates between two points on the line. This can also be determined by using the formula m = (y2 - y1) / (x2 - x1).

## What is the y-intercept of a line?

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form, b represents the y-intercept. In the point-slope form, the y-intercept can be found by substituting 0 for x and solving for y.

## How do I write the equation of a line given two points?

To write the equation of a line given two points, you can use the point-slope form. Substitute the coordinates of one point for (x1, y1) and the slope calculated from the two points for m. Then, solve for y to get the equation in point-slope form.

## What is the difference between a positive and negative slope?

A positive slope means that as the x-coordinate increases, the y-coordinate also increases. A negative slope means that as the x-coordinate increases, the y-coordinate decreases. A slope of 0 means that the line is horizontal, and a slope of undefined means that the line is vertical.

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