Explanation:
To write an equation in standard form of a line with integer intercepts, we can use the fact that the intercepts occur when the value of either the x-coordinate or the y-coordinate is zero.
Let's assume the x-intercept is at (a, 0) and the y-intercept is at (0, b), where a and b are integers.
The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
To find the slope, we can use the formula: m = (y2 - y1) / (x2 - x1), using the coordinates of the intercepts.
For the x-intercept (a, 0), the slope is: m = (0 - b) / (a - 0) = -b / a.
For the y-intercept (0, b), the slope is: m = (b - 0) / (0 - a) = b / (-a).
Since the intercepts are integers, it means that a and b are also integers.
Now, we can substitute the slope and the y-intercept (0, b) into the slope-intercept form equation:
y = (b / (-a))x + b.
To convert the equation to standard form, we multiply both sides of the equation by -a to eliminate the fraction:
-a * y = bx + (-a) * b,
-a * y - bx = -ab.
Finally, we rearrange the terms to get the equation in standard form:
ax + by = -ab.
In this equation, the intercepts are integers since a and b are integers.
To summarize, we can write an equation in standard form of a line with integer intercepts as ax + by = -ab, where a and b are integers representing the x-intercept and y-intercept, respectively.