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{SECT 0 {PARA 0 "" 0 "" {TEXT 256 49 "High School Modules > Algebra by
 Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 3 "   " }{TEXT 259 22 "L
ines in Standard Form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 138 "An exploration of lines in standard, Ax + By = C form
 - graphs, intercepts, slopes, converson to slope-intercept form, and \+
other features." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 257 153 "[Directions : Execute the Code Resource section first. \+
Although there will be no output immediately, these definitions are us
ed later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "restart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 549 "InterceptPlot  :=  proc( expr )\n     local a, b, A,
 B, C, wid, xi, yi, xl, xr, yl, yr;\n     a := solve( subs( y = 0, exp
r), x):\n     b := solve( subs( x = 0, expr), y):\n     xl := -abs(a*1
.3); xr := abs(a*1.3);\n     yl := -abs(b*1.3); yr := abs(b*1.3); \n  \+
   if ( a < 0 ) then xr := xr/3; else xl := xl/3; fi;\n     if ( b < 0
 ) then yr := yr/3; else yl := yl/3; fi;\n     wid := a/30;\n     xi :
= plottools[disk]([ a, 0], wid, color=red ):\n     yi := plottools[dis
k]([ 0, b], wid, color=green ):\n     display( xi, yi, scaling=constra
ined);\n end proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 664 "St
andardPlot  :=  proc( expr )\n     local a, b, A, B, C, wid, xi, yi, x
l, xr, yl, yr;\n     a := solve( subs( y = 0, expr), x):\n     b := so
lve( subs( x = 0, expr), y):\n     xl := -abs(a*1.3); xr := abs(a*1.3)
;\n     yl := -abs(b*1.3); yr := abs(b*1.3); \n     if ( a < 0 ) then \+
xr := xr/3; else xl := xl/3; fi;\n     if ( b < 0 ) then yr := yr/3; e
lse yl := yl/3; fi;\n     wid := a/30;\n     A := plot( solve( expr, y
), x = xl..xr, y=yl..yr, \n                           thickness = 2, c
olor = blue ):\n     xi := plottools[disk]([ a, 0], wid, color=red ):
\n     yi := plottools[disk]([ 0, b], wid, color=green ):\n     displa
y( A, xi, yi, scaling=constrained);\n end proc:" }}}}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 34 "1. Graphing Lines in Standard Form" }}{PARA 0 "" 
0 "" {TEXT -1 235 "Every equation of the form Ax + By = C represents a
 line. Assuming A and B are both non-zero, this kind of equation would
 always represent a non-horizonatal and non-vertical line. These lines
 ALWAYS have an x intercept and y intercept." }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 52 "implicitplot( 3*x + 4*y = 12,x = -5..5, y = -5..5 \+
);" }}}{PARA 0 "" 0 "" {TEXT -1 72 "\nNote that A, B, or C can be nega
tive also - or any combination of them." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "implicitplot( 3*x -7*y = 12,x = -5..5, y = -5..5 );" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
27 "2. Finding x & y intercepts" }}{PARA 0 "" 0 "" {TEXT -1 158 "\n It
s not too hard to find the x-intercept and y-intercept for a line in s
tandard form. To find the x-intercept there are two steps :\n    1. Su
bstitute y = 0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "9*x - 2*y
 = -36;\nsubs( y = 0, % );" }}}{PARA 0 "" 0 "" {TEXT -1 19 "\n    2. S
olve for x" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "XI = solve( %,
 x );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 
"\nThats the x intercept! We can find the y-intercept in a very simila
r way:\n\n    1. Substitute x = 0" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "9*x - 2*y = -36;\nsubs( x = 0, % );" }}}{PARA 0 "" 0 
"" {TEXT -1 18 "    2. Solve for y" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "YI = solve( %, y );" }}}{PARA 0 "" 0 "" {TEXT -1 29 "
\n\nHere are some more examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 114 "line := \n          -37*x + 74*y = -37;\nXI = solve( subs( y \+
= 0, line ), x) ;\nYI = solve( subs( x = 0, line ), y) ;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "line := \n          100*x + 25*y =
 125;\nXI = solve( subs( y = 0, line ), x) ;\nYI = solve( subs( x = 0,
 line ), y) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "line := \+
\n          .002*x + .006*y = 300;\nXI = solve( subs( y = 0, line ), x
) ;\nYI = solve( subs( x = 0, line ), y) ;" }}}}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 44 "3. Method of Graphing lines in Standard Form" }}{PARA 
0 "" 0 "" {TEXT -1 156 "\nWhen you are asked to graph lines in Standar
d form, you need to find the x-intercept and y-intercept then draw a l
ine connecting the dots to form the line." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "InterceptPlot( 3*x + 4*y = 12 );" }}}{PARA 0 "" 0 "" 
{TEXT -1 110 "\nNow we draw the line through these two points. There i
s only one unique line which passes through two points." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "StandardPlot( 3*x + 4*y = 12 );" }}
}{PARA 0 "" 0 "" {TEXT -1 109 "\nWhen graphing these lines, the best a
nd fastest method is to find the intercepts and then 'connect the dots
'" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "line := \n           14
*x - 21*y = 28;\nInterceptPlot( line );\nStandardPlot( line );" }}}
{PARA 0 "" 0 "" {TEXT -1 26 "\nHere are other examples. " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "line := \n           200*x - 50*y =
 1000;\nInterceptPlot( line );\nStandardPlot( line );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "line := \n           .02*x - .01*y \+
= -1;\nInterceptPlot( line );\nStandardPlot( line );" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 84 "line := \n           80*x + 60*y = -240;
\nInterceptPlot( line );\nStandardPlot( line );" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 47 "4. Finding the S
lope of a line in Standard Form" }}{PARA 0 "" 0 "" {TEXT -1 69 " \nGiv
en a line in standard form, we can find a formula for the slope." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "line := A*x + B*y = C;" }}}
{PARA 0 "" 0 "" {TEXT -1 68 "\nNow lets get rid of the smaller x term,
 so only one x term remains." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "m = solve( subs( \{x = 1, C = 0\}, line ), y );" }}}{PARA 0 "" 0 "
" {TEXT -1 24 "\nHere are some examples." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "line := 3*x + 5*y = 7;\nsolve( line, y ):  m = subs( \+
x = 1, % - subs( x = 0, %));\n\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "line := 8*x - 2*y = 37;\nsolve( line, y ):  m = subs(
 x = 1, % - subs( x = 0, %));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 87 "line := 100*x + 50*y = 20000;\nsolve( line, y ):  m = subs( x = \+
1, % - subs( x = 0, %));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 
"line := .0036*x + .144*y = 1;\nsolve( line, y ):  m = subs( x = 1, % \+
- subs( x = 0, %));" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 37 "5. Converting to Slope-Intercept Form" }}
{PARA 0 "" 0 "" {TEXT -1 58 "\nWe can convert to slope-intercept form \+
by solving for y. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Standa
rd_line := 3*x + 4*y = -12;\ny = solve(  %, y);" }}}{PARA 0 "" 0 "" 
{TEXT -1 53 "\nNote that we get the same graph, no matter the form." }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot( solve(Standard_line, \+
y), x = -5..5, thickness = 2, color = green);" }}}{PARA 0 "" 0 "" 
{TEXT -1 112 "\nHere are other examples. As long as A and B are non-ze
ro, we can also express the line in slope-intercept form." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "550*x + 275*y = 1;\ny = solve(  %, \+
y);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "-3000*x + 400*y = \+
1200;\ny = solve(  %, y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "72*x + 144*y = -360;\ny = solve(  %, y);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "sqrt(2)*x - sqrt(2)*y = 2;\ny = solve(  %, y);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x - .0008*y = -2;\ny = so
lve(  %, y);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 57 "6. Comparing St
andard Form Lines with Different Constants" }}{PARA 0 "" 0 "" {TEXT 
-1 108 "\nSuppose we have two lines in standard form which look identi
tal except for the constant term. For example :" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 74 "line1 :=\n          7*x - 8*y = 200;\nline2 :=
\n          7*x - 8*y = -55;\n\n\n" }}}{PARA 0 "" 0 "" {TEXT -1 77 "\n
What is the connection between them? We can get some clues by graphing
 them." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "display(StandardPl
ot(line1), StandardPlot(line2) );" }}}{PARA 0 "" 0 "" {TEXT -1 44 "\nI
nteresting. Let's try some other examples." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 95 "leftside := 4*x + 5*y:\ndisplay( StandardPlot( left
side = 10), StandardPlot( leftside = -10) );\n" }}}{PARA 0 "" 0 "" 
{TEXT -1 71 "\nIt looks like the lines are parallel. Lets explore this
 in more depth." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "leftside \+
:= 4*x + 5*y:\ndisplay( seq( StandardPlot( leftside = 5*k), k = 1..7) \+
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "leftside := -x + 4*y
:\ndisplay( seq( StandardPlot( leftside = 5*k), k = 1..7),\n         s
eq( StandardPlot( leftside = -5*k), k = 1..7) );" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 87 "\nYes, they all seem
 to be parallel. Lets check out this theory by comparing the slopes." 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "leftside := -x + 4*y:\nli
ne1 :=  leftside = -209;\nline2 :=  leftside = -13;\nline3 :=  leftsid
e = 55;\nline4 :=  leftside = 333;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 239 "solve( line1, y ):  m1 = subs( x = 1, % - subs( x = \+
0, %));\nsolve( line2, y ):  m1 = subs( x = 1, % - subs( x = 0, %));\n
solve( line3, y ):  m1 = subs( x = 1, % - subs( x = 0, %));\nsolve( li
ne4, y ):  m1 = subs( x = 1, % - subs( x = 0, %));" }}}{PARA 0 "" 0 "
" {TEXT -1 154 "\nThey all have the same slope. We can therefore concl
ude that the constant term has no affect on the slope. And lines with \+
the same A and B, are parallel." }}}{PARA 0 "" 0 "" {TEXT 258 36 "\n  \+
       \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
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