A line is shown on the coordinate plane.
On coordinate plane, points (2, 1), (1, 5) and (0, 9) are connected through a line.
Drag a number to each box to represent the equation of the line.
The equation of the line through points (2, 1), (1, 5), and (0, 9) is:
y = -4x + 9
To determine the equation of the line passing through the points (2, 1), (1, 5), and (0, 9), we 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 a point on the line, and m is the slope of the line.
Step 1: Calculate the slope (m) using any two of the given points. Let's choose the points (2, 1) and (1, 5).
m = (y₂ - y₁) / (x₂ - x₁)
= (5 - 1) / (1 - 2)
= 4 / -1
= -4
Step 2: Choose one of the given points as (x₁, y₁) and substitute the values into the point-slope equation. Let's use the point (2, 1) as (x₁, y₁).
y - 1 = -4(x - 2)
Step 3: Simplify the equation.
y - 1 = -4x + 8
Step 4: Rearrange the equation to isolate y.
y = -4x + 8 + 1
= -4x + 9
Therefore, the equation of the line passing through the points (2, 1), (1, 5), and (0, 9) is y = -4x + 9.
To find the equation of a line, we can use the slope-intercept form, which is given by:
y = mx + b
where:
- m represents the slope of the line
- b represents the y-intercept of the line
To determine the values of m and b, we can use two points on the line. Let's choose the points (1, 5) and (0, 9):
Step 1: Calculate the slope
The formula for slope (m) is given by:
m = (y2 - y1) / (x2 - x1)
Using the points (1, 5) and (0, 9), we can substitute the values into the formula:
m = (9 - 5) / (0 - 1)
m = 4 / (-1)
m = -4
So, the slope of the line is -4.
Step 2: Calculate the y-intercept
Now, we need to find the value of b. We can choose any of the given points to substitute into the equation y = mx + b. Let's use the point (2, 1):
1 = -4(2) + b
1 = -8 + b
b = 1 + 8
b = 9
So, the y-intercept of the line is 9.
Now we can write the equation of the line using the values we found:
y = -4x + 9
Thus, the equation of the line is y = -4x + 9.