Exercise 16 Page 214

Let's use the following fact: if the slopes of two distinct nonvertical lines are equal, then the lines are parallel. To find the y-coordinate of the point (6,y), for which the lines AB and CD are parallel, we will calculate the slopes of the lines and equate them.

Slope of AB

We can calculate the slope of the line AB using the Slope Formula. m = y_2-y_1/x_2-x_1 Let's use the coordinates of the first point (- 2,- 4) for (x_1,y_1), and the coordinates of the second point (6,8) for (x_2,y_2).

m=y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( - 2,- 4) & ( 6,8)

m=8-( - 4)/6-( - 2)

SubNeg

a-(- b)=a+b

m=8+4/6+2

AddTerms

Add terms

m=12/8

ReduceFrac

a/b=.a /4./.b /4.

m=3/2

The slope of line AB is 32.

Slope of CD

Similarly, we can calculate the slope of the line CD. Let's substititute (6,y) for (x_1,y_1) and (12,10) for (x_2,y_2) into the Slope Formula.

m=y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 6,y) & ( 12,10)

m=10-( y)/12-( 6)

SubTerm

Subtract term

m=10-y/6

The slope of line CD is 10-y6, where y is the unknown that we want to find.

Equate the Slopes

As we mentioned before, two lines are parallel if their slopes are equal. Therefore, in order for the lines AB and CD to be parallel, their slopes should be the same. Let's equate them! 3/2=10-y/6 If we solve this equation we will find the value of y when the lines are parallel.

3/2=10-y/6

CrossMult

Cross multiply

18=2(10-y)

Distr

Distribute 2

18=20-2y

AddEqn

LHS+2y=RHS+2y

2y+18=20

SubEqn

LHS-18=RHS-18

2y=2

DivEqn

.LHS /2.=.RHS /2.

y=1

The lines are parallel if y is equal to 1.

Check the Answer

Let's now plot the points, draw the lines through them, and see if they are really parallel.

As we can see, the lines are indeed parallel. The value of y is calculated correctly.