How does an equation of a line work?

Question

I realise there are questions asked similar to this, but I am still confused on the subject. It would be extremely helpful if explanations could be explained in-depth, but I could still understand what is being said.

How do equations in the form "y = mx + b" work, where this represents a straight line with slope m and y-intersect b?

Some example questions are listed. Not all of them have to be explained (though it would be appreciated).
1. State the slope and the y-intersect for "y = 3x + 5"
2. Write the equation of each line with the given slope of m = 2 and y-intersect of b = 3.
3. State the slope, y-intersect, and the equation for a line (a line on a graph is given with a right triangle along the line with the hypotenuse on the line, with the endpoints (0,1) and (2,2).
4. Determine the equation of a line (a line on a graph is given with two points, one at (-1,3) and the other at (3,-1).

Thanks for your time!

Answer 1

1. In this problem, all you have to do is state the 'm' and the 'b'. The slope is 'm' and the y intercept is the 'b'
2. This one is the exact opposite as the previos one. All you have to do is write the equation "y=mx+b" but replace the 'm' with the slope and replace the 'b' with th y intercept. So the answer is "y=2x+3"
3. The slope is alwise 'rise over run'. This means that you take the diference in the y value over the difference in the x.

The two y values are 1 and 2. 2-1=1, so the difference is 1. The two x values are 0 and 2. 2-0=2, so the difference is 2.

Now that you have your two differences, put them into a fraction. The slope is 1/2 (difference in x/difference in y). This means you go up one and right 2. (if the slope is 4/3 you go up 4 and right 3)
to extend your line the other way you go down 1 and left 2.

Now that you have the slope, draw the line. Start at one of the endpoints. Lets say you chose (0,1) Since the slope is 1/2, you go up 1 and right 2. When you do this, you get to the point (2,2)
If you do this again, you would get to the point (4,3) then (6,4) and so on.
Do this the other way now, starting again from (0,1). Go down one and left 2, and you get to the point (-2,0) then (-4,-1) Now that your line is extended, look at where the line crosses the y axis. This point is (0,1). The y value in this point is one, so your y intercept is 1

Since you have your slope (1/2) and y intercept (1), you can write your equation. Do what you did for number 2. replace 'm' with your slope and 'b' with your y intercept for the equation y=mx+b.

The equation for your line is "y=1/2x+1

4. For this one, do exactly what you did for the previos question. Find your difference in your y value ( 3+1, since it is a negative) and your difference in your x value (3+1 also), and you get 4/4, which is 1. Your slope is 1, or 1/1. Now extend your line by going up 1, right 1, or by going down 1, left 1, until your line crosses the y axis.


I hope this helped.

Answer 2

Okay, first off
The equation y =mx+b is just a formula to help plot a line on a graph where you know either the slope and y-intercept, the slope and one other point( y-y1)=m(x-x1) where the point (x1,y1) is not the y-int., or you just know two points on the line (x1,y1),(x2,y2).

1. y=mx+b y=3x+5 look familiar? What is m in the 2nd equation? What is b?
2. same as #1 only you are given m and b
3. A little trickier, but not too bad. You are actually given b in the problem (0,1) remember b = y-intercept or point where line crosses y-axis. What you need to find is the slope of the line. The slope is the change in y divided by change in x or, rise over run. The formula is (y2-y1)/(x2-x1). In this case, (2-1)/(0-1), which gives a slope of 1/2. So, m= 1/2 and b = what? (Remember, the y-intercept is (0,1) which means that b = 1.
4. Follow the same steps above to find the slope and then use the formula (y-y1) = m(x-x1) As a general rule, I always assign (x1,y1) as the first point given in a problem, which in this case is (-1,3).
Hope this helps!

Answer 3

Using your example, the number before the x, (3) is the slope of the line. At the point the x=0, y is equal to the constant/intercept; (5).

For questions 3 & 4, you need to determine the slope and intercept of the line (rise/run) and find the intercept when x=0.

Good luck

Answer 4

Slope intercept for is: y = mx + b on the grounds which you have each and all of the values different than b you are able to fill in and sparkling up for b. y = mx + b -5 = (4/5)(0) + b -5 = 0 + b b = -5 So, y = (4/5)x - 5