<p>1) At a certain high school, there are 9 students on the math team and 10 students on the academic team. There are a total of 11 students who are on exactly one of the teams. Of those students who are on the math team, how many are also on the academic team?</p>
<p>I can do it logically, but how do you do it using an algebraic set up?</p>
<p>2) IN the xy plane, the line with equation y=8 - 4/3x intersects the x-axis and y-axis at the points A and Bm respectively. What is the length of the line segment AB?</p>
<p>1#
You should think of the problem as a Venn diagram (which you might’ve done, since you solved it logically). So, 9 students in one circle, and 10 students in the other. If 11 students are on exactly one team, 8 (= 19 (total) - 11) students are in both teams. Number of students in the math team: </p>
<p>8 + x = 9
8 + y = 10</p>
<p>x = 1
y = 2</p>
<p>2#
y-intercept: when x = 0 ==> y = 8
x-intercept: when y = 0 ==> x = 6
Coordinates are: (0, 8) and (6, 0). You could use the Pythagorean theorem/distance formula to find the distance between the points.</p>
<p>But logic beats algebra: you have 19 slots to fill. 11 are filled by “singletons”, leaving 8 slots to be filled by kids who double up. So there are 4 of those kids.</p>
<p>This is a common SAT theme – good to be able to do it without to much fuss.</p>
<p>And, fwiw, there is a “visual” way to solve this problem. Of course, it is similar to the other solutions and uses logic. </p>
<p>Here we go.</p>
<p>If the groups are 9 and 10, we can look at the given numbers of 11 and distribute 11. The most logical manner is to use 5 and 6. Logical because the difference between the groups is only 1 and that we know that the “doublers” will not change the equation. </p>
<p>How many does it take to get from 5 to 9? Or from 6 to 10? </p>
<p>Of course, that is four. And, it is good to know that once a number works, we do not need to look further.</p>