A line with a slope of -2 passes through the point (1, 7). What is the closest point on the line above (1, 7) which has whole number coordinates?
You know that y decreases by 2 when x increases by 1.
So, (1,7) + (1,-2) = (2,5)
you could also go the other direction the same distance.
for a slope of -2
The y value would increase by 2 and the x value would decrease by 1. This gives us (0,9)
or
The y value would decrease by 2 and the x value would increase by 1 This gives us (2,5)
To find the equation of the line with a slope of -2 passing through the point (1, 7), we can use the point-slope formula.
The point-slope formula is given by:
y - y₁ = m(x - x₁)
Where (x₁, y₁) is the given point, m is the slope, and (x, y) represents any point on the line.
Substituting the values, we have:
y - 7 = -2(x - 1)
Simplifying the equation,
y - 7 = -2x + 2
y = -2x + 9
Now, to find the closest point on the line with whole number coordinates above (1, 7), we need to substitute different integer values for x and calculate the corresponding y-values.
Starting with x = 1, we find:
y = -2(1) + 9
y = -2 + 9
y = 7
The point (1, 7) lies on the line, but we need to find the next point above it.
Trying x = 2, we have:
y = -2(2) + 9
y = -4 + 9
y = 5
Therefore, the closest point on the line above (1, 7) with whole number coordinates is (2, 5).
To find the closest point with whole number coordinates on a line with a given slope that passes through a point, you can use the point-slope form of a linear equation.
Step 1: Start with the point-slope form of a linear equation, which is y - y1 = m(x - x1).
Given slope (m) = -2 and point (x1, y1) = (1, 7), we can substitute these values into the equation:
y - 7 = -2(x - 1)
Step 2: Simplify the equation by distributing -2 to (x - 1).
y - 7 = -2x + 2
Step 3: Move the terms around to isolate y.
y = -2x + 9
Now we have the equation of the line: y = -2x + 9.
Step 4: To find the closest point above (1, 7) with whole number coordinates, we need to examine the y-values for each whole number x-value greater than 1.
Let's start with x = 2:
y = -2(2) + 9
y = -4 + 9
y = 5
The point (2, 5) is on the line and above (1, 7). However, the y-value is not a whole number.
Next, let's try x = 3:
y = -2(3) + 9
y = -6 + 9
y = 3
The point (3, 3) is another point on the line, and the y-value is a whole number. It is also above (1, 7).
Therefore, the closest point on the line above (1, 7) with whole number coordinates is (3, 3).