Let 𝐴 be the point five, negative one and 𝐵 be the point negative one, eight. Which of the following points is on line 𝐴𝐵?
One thing we can do is find the function for this line and then plug in all of our options to see which of them makes the statement true. Since we’re given two points, the way that we can find the slope is by using the slope formula 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. We’ll take our two points and label them 𝑥 one, 𝑦 one; 𝑥 two, 𝑦 two. Plugging that in, we get eight minus negative one over negative one minus five, which equals nine over negative six.
If we divide the numerator and the denominator by three, we get negative three over two, which is the slope of this line. We have the slope of negative three-halves. But we’re still missing the 𝑦-intercept. To find 𝐵, we can use either one of our two points and plug them in. Using 𝑥 one, 𝑦 one — five, negative one — we now have the equation negative one equals negative three-halves times five plus b.
Five times negative three-halves is negative fifteen halves. And then, we need to add fifteen halves to both sides. On the right, they cancel out. On the left, I want to rewrite negative one as negative two over two so that we can add these two fractions together. Negative two over two plus 15 over two is thirteen halves. And that means our 𝑦-intercept is thirteen halves.
The equation for the line that both of these points fall on is 𝑦 equals negative three-halves 𝑥 plus 13 over two. We can plug in 𝑥 and 𝑦 for any point that falls on this line and the statement will be true. 𝑦 will equal negative three-halves times 𝑥 plus thirteen halves.
Now we just need to consider the options. Is negative seven equal to negative three-halves times three plus 13 over two? Starting with our first option, negative three-halves times three is negative nine-halves plus 13 over two equals four-halves which is two. Negative seven is not equal to two. And that means point a does not fall on our line.
Point b negative three-halves times seven equals negative twenty-one halves. Negative twenty-one halves plus thirteen halves equals negative eight-halves or just negative four. Seven is not equal to negative four. The point seven, seven does not fall on this line.
Point c negative three-halves times nine equals negative twenty-seven halves plus thirteen halves equals negative fourteen halves, which is negative seven. Negative seven is equal to negative seven. And that means the point nine, negative seven does fall on the line 𝐴𝐵.
Let’s quickly check the last two. Negative three-halves times negative seven equals positive twenty-one halves plus thirteen halves equals thirty-four halves, which is 17. Three is not equal to 17. Negative seven, three does not fall on this line.
And the Last option negative three-halves times negative seven equals twenty-one halves plus thirteen halves equals thirty-four halves, which again is 17. Nine is not equal to 17. And negative seven, nine does not fall on the line 𝐴𝐵.
Of these five choices, only the point nine, negative seven falls on the line 𝐴𝐵.