document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=708695>Question 708695</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What are 5 points that satisfy the equation 5y+3x=-13 ?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #436221 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=436221\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In theory, you could give either variable a value and solve for the other variable to get one point.<BR>\n" );
document.write( "You would do it again for each additional point.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Or you could solve for one variable as a function of the other,<BR>\n" );
document.write( "and get a formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%2B3x=-13 ALIGN=MIDDLE  ALT=\"5y%2B3x=-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y=-3x-13 ALIGN=MIDDLE  ALT=\"5y=-3x-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283x%2B13%29%2F5 ALIGN=MIDDLE  ALT=\"y=-%283x%2B13%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For any value of x, that formula makes it easy to calculate the corresponding value for y.<BR>\n" );
document.write( "Examples:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2A0%2B13%29%2F5 ALIGN=MIDDLE  ALT=\"y=-%283%2A0%2B13%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-13%2F5 ALIGN=MIDDLE  ALT=\"y=-13%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you point (0,-13/5)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2A1%2B13%29%2F5 ALIGN=MIDDLE  ALT=\"y=-%283%2A1%2B13%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2B13%29%2F5 ALIGN=MIDDLE  ALT=\"y=-%283%2B13%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-16%2F5 ALIGN=MIDDLE  ALT=\"y=-16%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you point (0,-16/5)<BR>\n" );
document.write( " <BR>\n" );
document.write( "However, for this equation, with those coefficients in front of x and y,<BR>\n" );
document.write( "you will end up with a lot of fractions, and may not like the results,<BR>\n" );
document.write( "especially if you need to use those points for a graph.<BR>\n" );
document.write( "If you want an integer value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > you need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B13 ALIGN=MIDDLE  ALT=\"3x%2B13\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a multiple of 5,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B13=3x%2B3%2B10=3%28x%2B1%29%2B5%2A2 ALIGN=MIDDLE  ALT=\"3x%2B13=3x%2B3%2B10=3%28x%2B1%29%2B5%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > you would want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1 ALIGN=MIDDLE  ALT=\"x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a multiple of 5,<BR>\n" );
document.write( "like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=5 ALIGN=MIDDLE  ALT=\"x%2B1=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2A4%2B13%29%2F5 ALIGN=MIDDLE  ALT=\"y=-%283%2A4%2B13%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%2812%2B13%29%2F5 ALIGN=MIDDLE  ALT=\"y=-%2812%2B13%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-25%2F5 ALIGN=MIDDLE  ALT=\"y=-25%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5 ALIGN=MIDDLE  ALT=\"y=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you point (4,-5).<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you do not need to \"show your work\" you could get additional points easily using a graph,<BR>\n" );
document.write( "without further calculations.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you need or want to keep calculating,<BR>\n" );
document.write( "increasing or decreasing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > by 5 you will keep getting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1 ALIGN=MIDDLE  ALT=\"x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a multiple of 5,<BR>\n" );
document.write( "and will keep getting integers for y.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you (9,-8)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you (-1,-2)<BR>\n" );
document.write( " <BR>\n" );
document.write( "GETTING ADDITIONAL POINTS FROM A GRAPH:<BR>\n" );
document.write( "Once you have one point of the line, you can plot it on a graph,<BR>\n" );
document.write( "and then you can use the slope of the line to place more points on the graph.<BR>\n" );
document.write( "(If you do not need your graph, you may be able to get the same result adding numbers in your head,<BR>\n" );
document.write( "or counting on your fingers).<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%2B3x=-13 ALIGN=MIDDLE  ALT=\"5y%2B3x=-13\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we could solve for y to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283x%2B13%29%2F5 ALIGN=MIDDLE  ALT=\"y=-%283x%2B13%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "which made it easy to calculate one point.<BR>\n" );
document.write( "We can go further to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28slope-intercept%29 ALIGN=MIDDLE  ALT=\"highlight%28slope-intercept%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > form of the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283x%2B13%29%2F5 ALIGN=MIDDLE  ALT=\"y=-%283x%2B13%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=%28-3%2F5%29x-13%2F5%29 ALIGN=MIDDLE  ALT=\"highlight%28y=%28-3%2F5%29x-13%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where the coefficient of the x is the slope of the line.<BR>\n" );
document.write( "Going from one point in the line to another point in the line,<BR>\n" );
document.write( "the slope is the change in y divided by the change in x.<BR>\n" );
document.write( "A slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F5 ALIGN=MIDDLE  ALT=\"-3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > means that<BR>\n" );
document.write( "an increase of 5 units in x corresponds to<BR>\n" );
document.write( "an increase of -3 units (meaning a decrease of 3 units) in y.<BR>\n" );
document.write( "We use that fact to go from one point to the next on the graph.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-8%2C17%2C-18%2C7%2C%0D%0Agrid%282%29%2C%0D%0Aline%28-11%2C4%2C19%2C-14%29%2Ccircle%28-6%2C1%2C0.3%29%2C%0D%0Acircle%2814%2C-11%2C0.3%29%2Ccircle%28-1%2C-2%2C0.3%29%2C%0D%0Ared%28circle%284%2C-5%2C0.4%29%29%2Ccircle%289%2C-8%2C0.3%29%2C%0D%0Alocate%282%2C-5.2%2C%22%284%2C-5%29%22%29%2C%0D%0Agreen%28arrow%284%2C-5%2C9%2C-5%29%29%2Cgreen%28arrow%289%2C-5%2C9%2C-7.8%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We move 5 units to the right (an increase of 5 for x)<BR>\n" );
document.write( "and 3 units down (decrease of 3 for y) to mark the next point.<BR>\n" );
document.write( "We can also reverse direction, decreasing x by 5, while increasing y by 3.<BR>\n" );
document.write( "So after calculating (4,-5) as a first point, we could go to (9,-8) and (14,-11).<BR>\n" );
document.write( "Reversing direction we could go to (-1,-2) and to (-6,1).<BR>\n" );
document.write( "Without graphing:<BR>\n" );
document.write( "I would just make a table for the x and y values and put the first point I calculate on the table.<BR>\n" );
document.write( "In this case, I would put x=4, y=-5 in the middle.<BR>\n" );
document.write( "Then I would fill the x values counting by 5's up and down,<BR>\n" );
document.write( "and the y values counting by -3 (counting by 3 in the other direction):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C100%2C1%2C26%2C-11%2C-1%2C%0D%0Aline%281%2C-6%2C26%2C-6%29%2Cline%284%2C-11%2C4%2C-1%29%2C%0D%0Alocate%282%2C-3%2Cx%29%2Clocate%286%2C-3%2C-6%29%2Clocate%2810%2C-3%2C-1%29%2C%0D%0Alocate%2815%2C-3%2C4%29%2Clocate%2819%2C-3%2C9%29%2Clocate%2823%2C-3%2C14%29%2C%0D%0Alocate%282%2C-8%2Cy%29%2Clocate%287%2C-8%2C1%29%2Clocate%2810%2C-8%2C-2%29%2C%0D%0Alocate%2814%2C-8%2C-5%29%2Clocate%2818%2C-8%2C-8%29%2Clocate%2822%2C-8%2C-11%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );