The tutor considers a problem most students encounter sometime….

When graphing a line on a page, fraction or decimal coordinates are inconvenient. An estimate is better than nothing, but one would like to avoid decimals or fractions when possible. In math, one often can. (In science labs, virtually everything is decimal anyway; estimates are expected.)

Example 1: Graph 2x + 7y= 13

Solution:

Setting, in turn, x and y to zero, we arrive at

Neither intercept is an integer. Can we perhaps find some other points on the line that are?

The key is to imagine combinations that add to 13. For this equation, 6+7=13 is particularly useful. Setting x to 3 and y to 1, we arrive at

2(3) + 7(1) = 13

There is, however, not much promise among the other sums. For example, there is no way to fit 5 + 8 = 13 to our equation without x or y being a fraction.

A combination in which one number is positive, but the other is negative, might work. Note there are infinitely many:

14-1=13
15-2=13
16-3=13
.
.
.
20-7=13

For our case, 20-7=13 is useful. We can fit it to our equation as follows:

2(10)+7(-1)=13

Here are the two points we have found that have integer coordinates:

2x + 7y= 13

When both the coefficients are even, but the number on the right is odd, this process will not yield a point with two integer coordinates, although it can likely give better coordinates than the intercepts. While not always successful, the process is more likely to be successful when the numbers seem awkward, which is when it’s perhaps needed most:)

HTH:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.