using the graph derive the equation of the line in slope-intercept form
To derive the equation of the line in slope-intercept form using the graph, follow these steps:
1. Identify two points on the line.
2. Find the slope of the line using the formula:
slope (m) = (change in y) / (change in x) = (y₂ - y₁) / (x₂ - x₁)
where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
3. Write the equation of the line using the slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept, the point where the line intersects the y-axis.
4. Substitute the slope (m) and one of the points' coordinates (x, y) into the equation to find the y-intercept (b).
5. Write the final equation.
Example:
Let's say we have a graph and we identify two points on the line: (2, 3) and (4, 5).
The slope (m) can be calculated as:
m = (y₂ - y₁) / (x₂ - x₁)
= (5 - 3) / (4 - 2)
= 2 / 2
= 1
So, the slope is 1.
Now, substitute the slope (m = 1) and one of the points' coordinates (x = 2, y = 3) into the equation y = mx + b to find the y-intercept (b):
3 = (1)(2) + b
3 = 2 + b
b = 3 - 2
b = 1
The y-intercept (b) is 1.
Finally, write the equation using the slope-intercept form:
y = mx + b
y = 1x + 1
y = x + 1
The equation of the line is y = x + 1.