Exercise 59 Page 327

We are given four equations in standard form in the same coordinate plane. x+4y=8 4x-y=-1 x+4y=- 12 4x-y=20 We want to draw their graphs and find out what figure these four lines form. We will do this by following two steps.

1. We will rewrite the equations in slope-intercept form.
2. We will graph the equations by using their y-intercepts and slopes.

   Equations in Slope-Intercept Form

   Let's begin by writing x+4y=8 in slope-intercept form.
   x+4y=8

   ▼

   Write in slope-intercept form

   SubEqn

   LHS-x=RHS-x

   4y=8-x

   DivEqn

   .LHS /4.=.RHS /4.

   y=8-x/4

   WriteDiffFrac

   Write as a difference of fractions

   y=8/4-x/4

   CalcQuot

   Calculate quotient

   y=2-x/4

   CommutativePropAdd

   Commutative Property of Addition

   y=- x/4+2

   MoveNumRight

   a/b=1/b* a

   y=- 1/4x+2
   With the same procedure we can write the other equations in slope-intercept form as well. We will make a table to write the other equations in this form.

   Equations in Standard From	Equations in Slope-Intercept Form
   x+4y=8	y=- 1/4x+2
   4x-y=-1	y=4x+1
   x+4y=- 12	y=- 1/4x-3
   4x-y=20	y=4x-20

   We are ready to draw the graphs of the equations!

   Drawing the Graphs

   We will draw y=- 14x+2 by using its y-intercept and its slope.

   

   With the same procedure we will draw the other lines.

   
   It appears as though it may be a square. However, we would need to check the slopes and side lengths to confirm this theory.

   Extra

   Further Information

   For further information we will check the slopes and side lengths. We will follow three steps.

   1. We will check the slope of these four lines.
   2. We will find the points of intersections.
   3. Then we will use these points to find the side lengths of the figure.

   We will begin by checking the slopes.

   Checking the Slopes

   To find the figure that four lines form we will use the slopes of the lines. Let's look at the slope-intercept form of the given equations and examine their slopes.

   Equations in Slope-Intercept Form	Slopes
   y=- 1/4x+2	- 1/4
   y=4x+1	4
   y=- 1/4x-3	- 1/4
   y=4x-20	4

   Recall that the slopes of perpendicular lines are opposite reciprocals, and the slopes of parallel lines are the same.

   

   Since all four angles of the quadrilateral are right angles, this figure is a rectangle. Now we will investigate whether the given figure is a square.

   Points of Intersections

   We will find each point of intersections. Let's begin with the intersection point of y=- 14x+2 and y=4x+1.

   y=- 14x+2 & (I) y=4x+1 & (II)

   ▼

   Solve by elimination

   MultEqn

   (I): LHS * 16=RHS* 16

   16y=-4x+32 y=4x+1

   SysEqnAdd

   (I): Add (II)

   16y+ y=-4x+32+ 4x+1 y=4x+1

   AddTerms

   (I): Add terms

   17y=33 y=4x+1

   DivEqn

   (I): .LHS /17.=.RHS /17.

   y= 3317 y=4x+1

   We found that y= 3317. We can substitute it into the second equation to find the value of x.

   y= 3317 y=4x+1

   Substitute

   (II): y= 33/17

   y= 3317 3317=4x+1

   ▼

   (II): Solve for x

   SubEqn

   (II): LHS-1=RHS-1

   y= 3317 3317-1=4x

   WriteAsFrac

   (II): Write as a fraction

   y= 3317 3317- 1717=4x

   SubFrac

   (II): Subtract fractions

   y= 3317 1617=4x

   DivEqn

   (II): .LHS /4.=.RHS /4.

   y= 3317 417=x

   RearrangeEqn

   (II): Rearrange equation

   y= 3317 x= 417

   We know that the intersection point of y=- 14x+2 and y=4x+1 is ( 417, 3317) . We can apply the same procedure to the other equation pairs to find their intersection points. We will make a table to show the intersection points.

   Pairs of Equations	Intersection Points
   y=- 1/4x+2 and y=4x+1	(4/17,33/17)
   y=- 1/4x+2 and y=4x-20	(88/17, 12/17)
   y=- 1/4x-3 and y=4x+1	(- 16/17,- 47/17)
   y=- 1/4x-3 and y=4x-20	(4,- 4)

   As we found the points of all vertices of the figure, now we can find the side lengths.

   Side Lengths of the Figure

   To find the side lengths of the figure we will use the Distance Formula. We will find the distance between ( 417, 3317) and ( 8817, 1217).

   d = sqrt((88/17-4/17)^2+(12/17-33/17)^2)

   ▼

   Simplify right-hand side

   SubFrac

   Subtract fractions

   d = sqrt((84/17)^2+(- 21/17)^2)

   CalcPow

   Calculate power

   d = sqrt((7056/289)+(441/289))

   SubTerm

   Subtract term

   d = sqrt(6615/289)

   UseCalc

   Use a calculator

   d = 5.09325 ...

   RoundDec

   Round to 2 decimal place(s)

   d ≈ 5.09

   Let's show all side lengths of the figure in a table.

   Points	Distance Formula	Side Lengths
   (4/17,33/17) and (88/17,12/17)	d = sqrt((88/17-4/17)^2+(12/17-33/17)^2)	≈ 5.09
   (- 16/17,- 47/17) and (4,- 4)	d = sqrt((- 16/17-4)^2+(- 47/17-(- 4))^2)	≈ 5.09
   (4/17,33/17) and (- 16/17,- 47/17)	d = sqrt((88/17-(- 16/17))^2+(12/17-(- 47/17))^2)	≈ 4.85
   (88/17,12/17) and (4,- 4)	d = sqrt((88/17-4)^2+(12/17-(- 4))^2)	≈ 4.85

   We will show the side lengths in the graph, as well.

   

   As all four sides of the quadrilateral are not equal, these figure is not a square. It is a rectangle.