The Correct Answer and Explanation is:
Of course. Based on the visible information in the image, the problem asks to find the equation of a line passing through the points (2, 5) and (4, 9).
Correct Answer:
y = 2x + 1
Explanation
The goal is to write the equation of a line in slope-intercept form, which is universally expressed as y = mx + b. In this formula, ‘m’ represents the slope of the line, and ‘b’ represents the y-intercept, which is the point where the line crosses the vertical y-axis. To find the complete equation, we must determine the values for both m and b using the two given points: (2, 5) and (4, 9).
Step 1: Calculate the Slope (m)
The slope measures the steepness of a line and is calculated as the “rise” (change in y) divided by the “run” (change in x). The formula for the slope using two points (x1, y1) and (x2, y2) is:
m = (y2 – y1) / (x2 – x1)
Let’s assign our points:
(x1, y1) = (2, 5)
(x2, y2) = (4, 9)
Now, we substitute these values into the formula:
m = (9 – 5) / (4 – 2)
m = 4 / 2
m = 2
So, the slope (m) of the line is 2. Our equation now looks like this: y = 2x + b.
Step 2: Find the Y-intercept (b)
With the slope known, we can find the y-intercept (b) by substituting the coordinates of one of the given points into our partial equation. Let’s use the point (2, 5), where x = 2 and y = 5.
y = 2x + b
5 = 2(2) + b
5 = 4 + b
To solve for b, we subtract 4 from both sides of the equation:
5 – 4 = b
1 = b
Thus, the y-intercept (b) is 1.
Step 3: Write the Final Equation
Now that we have calculated both the slope (m = 2) and the y-intercept (b = 1), we can write the final equation of the line by substituting these values back into the slope-intercept form:
y = 2x + 1
