document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173100>Question 1173100</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mr.Agoncillo has a savings account in two banks. The combined amount of these savings is at least Php 150,000. One bank gives an interest of 4% while the other bank give 6%. In a year, Mr. Agoncillo receive at most Php. 12,000. \r<BR>\n" );
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document.write( "Write and graph a system of linear inequalities that shows all the posible solutions. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850756 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850756\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem and set up the system of linear inequalities.\r<BR>\n" );
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document.write( "**Variables**\r<BR>\n" );
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document.write( "* Let 'x' represent the amount of savings in the bank with 4% interest.<BR>\n" );
document.write( "* Let 'y' represent the amount of savings in the bank with 6% interest.\r<BR>\n" );
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document.write( "**Inequalities**\r<BR>\n" );
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document.write( "1.  **Combined Savings:**<BR>\n" );
document.write( "    * The combined amount of savings is at least Php 150,000.<BR>\n" );
document.write( "    * Equation: x + y ≥ 150,000\r<BR>\n" );
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document.write( "2.  **Interest Earned:**<BR>\n" );
document.write( "    * The total interest earned is at most Php 12,000.<BR>\n" );
document.write( "    * Interest from 4% bank: 0.04x<BR>\n" );
document.write( "    * Interest from 6% bank: 0.06y<BR>\n" );
document.write( "    * Equation: 0.04x + 0.06y ≤ 12,000\r<BR>\n" );
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document.write( "3.  **Non-negative Savings:**<BR>\n" );
document.write( "    * Savings cannot be negative.<BR>\n" );
document.write( "    * Equations: x ≥ 0 and y ≥ 0\r<BR>\n" );
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document.write( "**System of Linear Inequalities**\r<BR>\n" );
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document.write( "* x + y ≥ 150,000<BR>\n" );
document.write( "* 0.04x + 0.06y ≤ 12,000<BR>\n" );
document.write( "* x ≥ 0<BR>\n" );
document.write( "* y ≥ 0\r<BR>\n" );
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document.write( "**Graphing the Inequalities**\r<BR>\n" );
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document.write( "1.  **x + y ≥ 150,000**\r<BR>\n" );
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document.write( "    * To graph this, first graph the line x + y = 150,000.<BR>\n" );
document.write( "    * Find the intercepts:<BR>\n" );
document.write( "        * If x = 0, y = 150,000<BR>\n" );
document.write( "        * If y = 0, x = 150,000<BR>\n" );
document.write( "    * Draw the line connecting (150000,0) and (0,150000). Since the inequality is \"greater than or equal to,\" use a solid line.<BR>\n" );
document.write( "    * Shade the region above the line, as that represents x + y ≥ 150,000.\r<BR>\n" );
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document.write( "2.  **0.04x + 0.06y ≤ 12,000**\r<BR>\n" );
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document.write( "    * To graph this, first graph the line 0.04x + 0.06y = 12,000.<BR>\n" );
document.write( "    * To make it easier to graph, we can multiply the entire equation by 100<BR>\n" );
document.write( "        * 4x+6y=1200000<BR>\n" );
document.write( "    * Then we can divide the entire equation by 2<BR>\n" );
document.write( "        * 2x+3y=600000<BR>\n" );
document.write( "    * Find the intercepts:<BR>\n" );
document.write( "        * If x = 0, 3y = 600000, y = 200,000<BR>\n" );
document.write( "        * If y = 0, 2x = 600000, x = 300,000<BR>\n" );
document.write( "    * Draw the line connecting (300000, 0) and (0, 200000). Since the inequality is \"less than or equal to,\" use a solid line.<BR>\n" );
document.write( "    * Shade the region below the line, as that represents 0.04x + 0.06y ≤ 12,000.\r<BR>\n" );
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document.write( "3.  **x ≥ 0 and y ≥ 0**\r<BR>\n" );
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document.write( "    * These inequalities restrict the solution to the first quadrant (where both x and y are non-negative).\r<BR>\n" );
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document.write( "**Finding the Solution Region**\r<BR>\n" );
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document.write( "* The solution region is the area where all shaded regions overlap in the first quadrant.<BR>\n" );
document.write( "* The corner points of this region are important for finding specific solutions.\r<BR>\n" );
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document.write( "**Corner Points**\r<BR>\n" );
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document.write( "* Intersection of x + y = 150,000 and x = 0: (0, 150,000)<BR>\n" );
document.write( "* Intersection of x + y = 150,000 and 0.04x + 0.06y = 12,000:<BR>\n" );
document.write( "    * Solve the system of equations.<BR>\n" );
document.write( "    * From x + y = 150,000, y = 150,000 - x.<BR>\n" );
document.write( "    * Substitute into 0.04x + 0.06(150,000 - x) = 12,000<BR>\n" );
document.write( "    * 0.04x + 9,000 - 0.06x = 12,000<BR>\n" );
document.write( "    * -0.02x = 3,000<BR>\n" );
document.write( "    * x = -150000. There is an error.<BR>\n" );
document.write( "    * multiply 0.04x + 0.06y = 12000 by 100. 4x+6y=1200000, then divide by 2. 2x+3y=600000<BR>\n" );
document.write( "    * multiply x+y=150000 by 2. 2x+2y=300000. Substract that from 2x+3y=600000. y=300000. x=-150000. There is still an error.<BR>\n" );
document.write( "    * Multiply x+y=150000 by 0.04. 0.04x+0.04y=6000.<BR>\n" );
document.write( "    * Subtract that from 0.04x+0.06y=12000. 0.02y=6000. y=300000. x=150000-300000. x=-150000. There is still an error.<BR>\n" );
document.write( "    * 0.04x + 0.06(150000-x)=12000<BR>\n" );
document.write( "    * 0.04x +9000-0.06x=12000<BR>\n" );
document.write( "    * -0.02x=3000<BR>\n" );
document.write( "    * x=-150000<BR>\n" );
document.write( "    * There is an error with the problem itself. The intersection of those two lines is outside of the first quadrant.\r<BR>\n" );
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document.write( "* Intersection of 0.04x + 0.06y = 12,000 and y = 0: (300,000, 0)<BR>\n" );
document.write( "* Intersection of x=0 and y=0 (0,0)\r<BR>\n" );
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document.write( "The solution region is the area between the lines, and within the first quadrant.<BR>\n" );
document.write( " </SPAN>\n" );
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