how to find th equation that passes though (1,3) and (8,5)
is there a formula to find the answer>
yes, there are several methods
first you need the slope
slope = (5-3)/(8-1) = 2/7
general form: y = mx + b, where m is the slope and b is the y-intercept
y = (2/7)x + b
but the point (1,3) is on it, so
3 = (2/7)(1) + b
multiply by 7 to get rid of fractions
21 = 2 + 7b
7b = 19
b = 19/7
then y = (2/7)x + 19/7
check by using the point not used in the last step, it must satisfy my equation
for (8,5)
5 = (2/7)(8) + 2/7
5 = 16/7 + 19/7
5 = 35/7
5=5
my equation is correct
a "nicer" way is to take the slope and a point and use the point-slope form
so m = 2/7, point (1,3)
y - 3 = (2/7)(x - 1)
again, times 7
7y - 21 = 2x - 2
2x - 7y = -19
Your algebra skills should be good enough to see that my two versions are the same
I should not have called y = mx + b the "general form"
y = mx + b is called the slope y-intercept form
general form would be: 2x - 7y + 19 = 0
some texts call 2x-7y=-19 the standard form
Yes, there is a formula to find the equation of a line that passes through two given points. The formula is called the point-slope form, which is given by:
y - y1 = m(x - x1)
Where:
- (x1, y1) are the coordinates of one of the given points on the line
- m is the slope of the line
To find the slope (m), you can use the following formula:
m = (y2 - y1) / (x2 - x1)
where (x2, y2) are the coordinates of the other given point on the line.
Let's apply these steps to the given points (1,3) and (8,5):
1. Find the slope (m):
m = (5 - 3) / (8 - 1)
m = 2 / 7
2. Choose one of the given points (let's use (1,3)) and substitute the values of x1, y1, and m into the point-slope form:
y - 3 = (2/7)(x - 1)
3. Simplify the equation:
y - 3 = (2/7)x - 2/7
4. Rearrange the equation to get the standard form:
(2/7)x - y = 2/7 - 3
2x - 7y = -17
Therefore, the equation of the line that passes through the points (1,3) and (8,5) is 2x - 7y = -17.