document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=20904>Question 20904</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Four equations are shown. Which lines are parallel? Which lines are perpendicular?<BR>\n" );
document.write( "a. y=-1/8x+2<BR>\n" );
document.write( "b. y=8x+4<BR>\n" );
document.write( "c. y=-8x+2<BR>\n" );
document.write( "d. 8x+y=4<BR>\n" );
document.write( "Fill in the blanks with the correct leters.<BR>\n" );
document.write( "Lines______ and ______ are parallel.<BR>\n" );
document.write( "Lines______ and ______ are perpendicular. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #12066 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=12066\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> LINES A AND B ARE PERPENDICULAR.<BR>\n" );
document.write( "LINES C AND D ARE PARALLEL.<BR>\n" );
document.write( "SEE BELOW FOR EXPLANATION<BR>\n" );
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document.write( "There is a line (L1) that passes through the points (8,-3) and (3,3/4)<BR>\n" );
document.write( "There is another line (L2)with slope M=2/3 that intersects L1 at the point<BR>\n" );
document.write( "(-4,6)<BR>\n" );
document.write( "What is the point of interectstion of line L1 and L2<BR>\n" );
document.write( "1 solutions<BR>\n" );
document.write( "Answer 8530 by venugopalramana(345) About Me on 2005-10-29 23:45:54 (Show Source):<BR>\n" );
document.write( "SEE THE FOLLOWING WHICH IS SIMILAR TO YOUR PROBLEM AND DO<BR>\n" );
document.write( "GIVEN:<BR>\n" );
document.write( "· There is a line (L1) that passes through the points<BR>\n" );
document.write( "(8,-3) and (3,3/4).<BR>\n" );
document.write( "· There is another line (L2) with slope m=2/3 that<BR>\n" );
document.write( "intersects L1 at the point (-4,6).<BR>\n" );
document.write( "· A third line (L3) is parallel to L2 that passes<BR>\n" );
document.write( "through the (7,-13 1/2).<BR>\n" );
document.write( "· Yet another line (L4) is perpendicular to L3, and<BR>\n" );
document.write( "passes through the point (1/2,5 2/3).<BR>\n" );
document.write( "· The fifth line (L5) has the equation<BR>\n" );
document.write( "2/5y-6/10x=24/5.<BR>\n" );
document.write( "Using whatever method, find the following:<BR>\n" );
document.write( "2. The point of intersection of L1 and L3<BR>\n" );
document.write( "3. The point of intersection of L1 and L4<BR>\n" );
document.write( "4. The point of intersection of L1 and L5<BR>\n" );
document.write( "5. The point of intersection of L2 and L3<BR>\n" );
document.write( "6. The point of intersection of L2 and L4<BR>\n" );
document.write( "7. The point of intersection of L2 and L5<BR>\n" );
document.write( "8. The point of intersection of L3 and L4<BR>\n" );
document.write( "9. The point of intersection of L3 and L5<BR>\n" );
document.write( "10. The point of intersection of L4 and L5<BR>\n" );
document.write( "PLEASE NOTE THE FOLLOWING FORMULAE FOR EQUATION OF A<BR>\n" );
document.write( "STRAIGHT LINE:<BR>\n" );
document.write( "slope(m)and intercept(c) form...y=mx+c<BR>\n" );
document.write( "point (x1,y1) and slope (m) form...y-y1=m(x-x1)<BR>\n" );
document.write( "two point (x1,y1)and(x2,y2)form.....................<BR>\n" );
document.write( "y-y1=((y2-y1)/(x2-x1))*(x-x1)<BR>\n" );
document.write( "standard linear form..ax+by+c=0..here by transforming<BR>\n" );
document.write( "we get by=-ax-c..or y=(-a/b)x+(-c/b)..comparing with<BR>\n" );
document.write( "slope intercept form we get ...slope = -a/b and<BR>\n" );
document.write( "intercept = -c/b<BR>\n" );
document.write( "*****************************************************<BR>\n" );
document.write( "line (L1) that passes through the points (8,-3) and<BR>\n" );
document.write( "(3,3/4).<BR>\n" );
document.write( "eqn.of L1..y-(-3)=((3/4+3)/(3-8))*(x-8)<BR>\n" );
document.write( "y+3=((15/4(-5)))(x-8)=(-3/4)(x-8)<BR>\n" );
document.write( "or multiplying with 4 throughout<BR>\n" );
document.write( "4y+12=-3x+24<BR>\n" );
document.write( "3x+4y+12-24=0<BR>\n" );
document.write( "3x+4y-12=0.........L1<BR>\n" );
document.write( "There is another line (L2) with slope m=2/3 that<BR>\n" );
document.write( "intersects L1 at the point (-4,6).<BR>\n" );
document.write( "This means (-4,6)lies on both L1 and L2.(you can check<BR>\n" );
document.write( "the eqn.of L1 we got by substituting this point in<BR>\n" );
document.write( "equation of L1 and see whether it is satisfied).So<BR>\n" );
document.write( "eqn.of L2...<BR>\n" );
document.write( "y-6=(2/3)(x+4)..multiplying with 3 throughout..<BR>\n" );
document.write( "3y-18=2x+8<BR>\n" );
document.write( "-2x+3y-26=0.........L2<BR>\n" );
document.write( "A third line (L3) is parallel to L2 that passes<BR>\n" );
document.write( "through the (7,-13 1/2).<BR>\n" );
document.write( "lines are parallel mean their slopes are same . so we<BR>\n" );
document.write( "keep coefficients of x and y same for both parallel<BR>\n" );
document.write( "lines and change the constant term only..<BR>\n" );
document.write( "eqn.of L2 from above is ...-2x+3y-26=0.........L2<BR>\n" );
document.write( "hence L3,its parallel will be ...-2x+3y+k=0..now it<BR>\n" );
document.write( "passes through (7,-13 1/2)=(7,-13.5)......substituting<BR>\n" );
document.write( "in L3..we get k<BR>\n" );
document.write( "-2*7+3*(-13.5)+k=0...or k=14+40.5=54.5..hence eqn.of<BR>\n" );
document.write( "L3 is........-2x+3y+54.5=0......................L3<BR>\n" );
document.write( "Yet another line (L4) is perpendicular to L3, and<BR>\n" );
document.write( "passes through the point (1/2,5 2/3).<BR>\n" );
document.write( "lines are perpendicular when the product of their<BR>\n" );
document.write( "slopes is equal to -1..so ,we interchange coefficients<BR>\n" );
document.write( "of x and y from the first line and insert a negative<BR>\n" );
document.write( "sign to one of them and then change the constant term.<BR>\n" );
document.write( "L3 is........-2x+3y+54.5=0......................L3<BR>\n" );
document.write( "hence L4,its perpendicular will be ..3x+2y+p=0...L4<BR>\n" );
document.write( "this passes through (1/2,5 2/3)=(1/2,17/3).hence..<BR>\n" );
document.write( "3*1/2+2*17/3+p=0..or ..p= -77/6.so eqn.of L4 is<BR>\n" );
document.write( "3x+2y-77/6=0..............L4<BR>\n" );
document.write( "The fifth line (L5) has the equation 2/5y-6/10x=24/5.<BR>\n" );
document.write( "now to find a point of intersection means to find a<BR>\n" );
document.write( "point say P(x,y) which lies on both the lines ..that<BR>\n" );
document.write( "is, it satisfies both the equations..so we have to<BR>\n" );
document.write( "simply solve the 2 equations of the 2 lines for x and<BR>\n" );
document.write( "y to get their point of intersection.For example to<BR>\n" );
document.write( "find the point of intersection of L1 and L3 we have to<BR>\n" );
document.write( "solve for x and y the 2 equations....<BR>\n" );
document.write( "3x+4y-12=0.........L1....(1) and<BR>\n" );
document.write( "-2x+3y+40.5=0......L3.....(2)<BR>\n" );
document.write( "I TRUST YOU CAN CONTINUE FROM HERE TO GET THE<BR>\n" );
document.write( "ANSWERS.If you have any doubts or get into any<BR>\n" );
document.write( "difficulty ,please ask me.<BR>\n" );
document.write( "venugopal \r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );