### Home > PC > Chapter 12 > Lesson 12.2.1 > Problem 12-42

12-42.

Recall from Chapter 1 that the line through the points

*A*and*B*can be described by the vector equationfor −∞ < *t*< ∞. Homework Help ✎Let

*A*= (2, 4) and*B*= (1, −3). Since= 2 + 4**i**and**j**= − **i**− 7**j**, verify that in this case+ *t*() = 2 **i**+ 4**j**+*t*(−**i**− 7**j**).Rewrite this expression as ( )i + ( )j.

Show that this line goes through the point (0, −10).

Use substitution.

Use the distributive property, then combine like terms.

2**i** + 4**j** + *t*(−**i** −7**j**)

= 2**i** + 4**j** −*t***i** −7*t***j**

= 2**i** −*t***i** + 4**j** −7*t***j**

If (0, 10) is on the line, then 0**i** + 10**j** is possible.

(2 − *t*)**i** + (4 − 7*t*)**j** = 0**i** + 10**j**

2 − *t* = 0 4 − 7*t* = 10

Does *t* have the same value in both equations?

If it is on the line, then it should.