write an equation for the line through the given points or through the given point with the given slope
(5,7),(6,8)
(1,2),(3,8)
(0,5),(-3,2)
(8,11),(6,16)
For each of them, use the standard formula
(y- y1) = [(y2-y1)/x2-x1)]*(x-x1)
In the first case, that gives you
(y-7) = (1/1)* (x-5)
y = x+2
drwls, thanks for the help. I was trying to check my sons answers. He got the first one right. When he is doing this homework he is not sure of what he is doing. I showed him this and asked if he understood the standard formula and he looked like a deer in headlights! I said to make the graph and maybe that would help. Any other help. Thanks
In the "standard formula"
[(y2-y1)/(x2-x1)]
is the slope of the line, usually called "m".
The standard formula is just is a mathematical statement that the change in y from a point (x1,y1) is equal to the slope, multiplied by the change in x measured from the same point. Some texts may use a different approach, but I find the one that I wrote easiest to remember.
Applying the same method to the second problem,
y -2 = (6/2) (x-1)
y = 2 + 3x -3
y = 3x - 1
Let's take a different approach to help your son understand the standard formula and graphing. Instead of relying solely on mathematical equations, let's use humor to make it more engaging and memorable for him. Here's a lighthearted explanation for each of the given points:
1. (5,7), (6,8)
Imagine a little snail named Snaily and his friend Turtle. Snaily lives at the point (5,7) and Turtle lives at (6,8). Since Snaily and Turtle are best friends, they decided to go on an adventure together and form a line. Snaily is very motivated, so he moves one step forward (x+1), and Turtle follows (y+1) to stay by Snaily's side. Hence, the equation of their journey is y = x + 1!
2. (1,2), (3,8)
Let's imagine two monkeys named Monkey A and Monkey B. Monkey A lives at point (1,2) and Monkey B lives at point (3,8). They are super competitive and always want to be one step ahead of each other. Monkey A decides to move two steps forward (x+2), leaving Monkey B far behind. Monkey B, feeling left out, decides to make up for it by climbing on a tree (y+6) to catch up to Monkey A. So, their equation is y = x + 6!
3. (0,5), (-3,2)
Imagine a penguin named Pengy and a walrus named Wally. Pengy lives at point (0,5) and Wally lives at point (-3,2). Pengy loves sliding on the ice and decides to slide three steps forward (x+3). Wally, seeing Pengy have so much fun, waddles three steps to catch up (y+3). Therefore, their equation is y = x + 3!
4. (8,11), (6,16)
Let's meet two crazy snakes named Slithery and Sly. Slithery lives at point (8,11) and Sly lives at (6,16). These sneaky snakes always try to outdo each other. Slithery decides to sneak two steps back (x-2), making Sly realize she needs to outshine him. Sly slithers five steps forward (y+5) to make sure she's ahead. So, their equation is y = x + 5!
Remember, humor can be a helpful tool in making learning enjoyable. Encourage your son to visualize the scenarios I've created for each equation, and he might find it easier to grasp the concept.
In the third problem, using the standard formula, we have:
(y - 5) = (2 - 5)/(-3 - 0)(x - 0)
(y - 5) = -1/3(x - 0)
y - 5 = -1/3x
y = -1/3x + 5
Lastly, for the fourth problem:
(y - 11) = (16 - 11)/(6 - 8)(x - 8)
(y - 11) = 5/-2(x - 8)
(y - 11) = -5/2(x - 8)
y - 11 = -5/2x + 20
y = -5/2x + 31/2
Sure! Let's go through the steps to find the equation for the line through the given points or through the given point with the given slope.
1) We'll use the standard formula:
(y - y1) = [(y2-y1)/(x2-x1)]*(x - x1)
2) Let's use the formula to find the equation for the line through the points (5,7) and (6,8):
We have:
(x1, y1) = (5,7)
(x2, y2) = (6,8)
Using the formula, we get:
(y - 7) = [(8-7)/(6-5)] * (x - 5)
Simplifying this gives us:
(y - 7) = (1/1) * (x - 5)
And finally, we can rewrite it as:
y = x + 2
So the equation for the line through the points (5,7) and (6,8) is y = x + 2.
3) Now let's find the equation for the line through the points (1,2) and (3,8):
We have:
(x1, y1) = (1,2)
(x2, y2) = (3,8)
Using the formula, we get:
(y - 2) = [(8-2)/(3-1)] * (x - 1)
Simplifying this gives us:
(y - 2) = (6/2) * (x - 1)
And finally, we can rewrite it as:
y = 3x - 1
So the equation for the line through the points (1,2) and (3,8) is y = 3x - 1.
4) Let's find the equation for the line through the points (0,5) and (-3,2):
We have:
(x1, y1) = (0,5)
(x2, y2) = (-3,2)
Using the formula, we get:
(y - 5) = [(2-5)/(-3-0)] * (x - 0)
Simplifying this gives us:
(y - 5) = (-3/-3) * x
And finally, we can rewrite it as:
y = x + 5
So the equation for the line through the points (0,5) and (-3,2) is y = x + 5.
5) Lastly, let's find the equation for the line through the points (8,11) and (6,16):
We have:
(x1, y1) = (8,11)
(x2, y2) = (6,16)
Using the formula, we get:
(y - 11) = [(16-11)/(6-8)] * (x - 8)
Simplifying this gives us:
(y - 11) = (5/-2) * (x - 8)
And finally, we can rewrite it as:
y = -5/2 * x + 34
So the equation for the line through the points (8,11) and (6,16) is y = -5/2 * x + 34.
I hope this helps! Let me know if you have any more questions.