To find the equation of a line that passes through the points (3, 4) and (2, 1), we will follow a systematic approach that involves determining the slope and then using it to find the y-intercept.
Step 1: Calculate the Slope (m)
The slope (m) of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
m = (y2 – y1) / (x2 – x1)
In this case, our points are (3, 4) and (2, 1). Assigning:
- (x1, y1) = (3, 4)
- (x2, y2) = (2, 1)
Now, substituting the values into the slope formula:
m = (1 – 4) / (2 – 3)
This simplifies to:
m = (-3) / (-1) = 3
Step 2: Use the Slope to Find the Equation
The slope-intercept form of a line is given by:
y = mx + b
Where:
- m = slope
- b = y-intercept
We now know that the slope (m) is 3. To find the y-intercept (b), we can substitute one of the points into the equation. Let’s use the point (3, 4):
4 = 3(3) + b
This simplifies to:
4 = 9 + b
Now, isolating b:
b = 4 – 9 = -5
Step 3: Write the Final Equation
y = 3x – 5
Conclusion
The equation of the line that passes through the points (3, 4) and (2, 1) in slope-intercept form is:
y = 3x – 5