If the slope of a line is 5/3 and a point on the line is (57,12) using the given point and the slope find the coordinates of the lattice point on the line?
The "point-slope" form of the equation of a straight line is:
y - y1 = m ( x - x1 )
(x1, y1) is a known point
m is the slope of the line
(x, y) is any other point on the line
In this case:
m = 5 / 3 , x1 = 57 , y1 = 12
y - 12 = ( 5 / 3 ) (x - 57 )
y = (5 x / 3 ) - ( 5 * 57 / 3 ) + 12
y = (5 x / 3 ) - ( 285 / 3 ) + 12
y = (5 x / 3 ) - 95 + 12
y = (5 x / 3 ) - 83
To find the coordinates of another lattice point on a line, given a point and the slope, you can use the point-slope form of a linear equation.
The point-slope form is given by:
y - y1 = m(x - x1),
where (x1, y1) is the given point on the line and m is the slope.
In this case, the slope is 5/3 and the given point is (57, 12).
Let's substitute these values into the point-slope form:
y - 12 = (5/3)(x - 57).
Now, we can simplify the equation:
3(y - 12) = 5(x - 57).
Expanding both sides of the equation:
3y - 36 = 5x - 285.
Rearranging the equation:
5x - 3y = 249.
Now, to find the lattice point, we need to plug in whole number values for x and y. Let's start by assigning a value to x. Suppose we let x = 0:
5(0) - 3y = 249.
Simplifying:
-3y = 249.
Dividing both sides by -3:
y = -83.
So, one lattice point on the line is (0, -83).
You can repeat this process by assigning different values to x, such as x = 1, x = 2, and so on, to find other lattice points on the line.