document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172925>Question 1172925</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mrs. Diez has 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 gives 6%. In a year, Mrs. Diez receives at most Php12,000.\r<BR>\n" );
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document.write( "a. What mathematical statements represent the given situation? <BR>\n" );
document.write( "b. Draw the graphs of the mathematical statements.<BR>\n" );
document.write( "c. How will you determine the amount of savings in each bank account? <BR>\n" );
document.write( "d. Give one possible amount of savings in both accounts.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850796 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=850796\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**a. Mathematical Statements**\r<BR>\n" );
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document.write( "Let:\r<BR>\n" );
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document.write( "* `x` = amount of savings in the bank with 4% interest<BR>\n" );
document.write( "* `y` = amount of savings in the bank with 6% interest\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( "    * Mathematical statement: `x + y ≥ 150000`\r<BR>\n" );
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document.write( "2.  **Total interest:**<BR>\n" );
document.write( "    * The total interest received is at most Php 12,000.<BR>\n" );
document.write( "    * Interest from the 4% bank: 0.04x<BR>\n" );
document.write( "    * Interest from the 6% bank: 0.06y<BR>\n" );
document.write( "    * Mathematical statement: `0.04x + 0.06y ≤ 12000`\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( "    * Mathematical statements: `x ≥ 0` and `y ≥ 0`\r<BR>\n" );
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document.write( "**b. Graphs of the Mathematical Statements**\r<BR>\n" );
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document.write( "1.  **x + y ≥ 150000:**<BR>\n" );
document.write( "    * To graph this, we first graph the line `x + y = 150000`.<BR>\n" );
document.write( "    * Find the intercepts:<BR>\n" );
document.write( "        * If x = 0, then y = 150000. (0, 150000)<BR>\n" );
document.write( "        * If y = 0, then x = 150000. (150000, 0)<BR>\n" );
document.write( "    * Draw the line connecting these points.<BR>\n" );
document.write( "    * Since it's `x + y ≥ 150000`, we shade the region above the line.\r<BR>\n" );
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document.write( "2.  **0.04x + 0.06y ≤ 12000:**<BR>\n" );
document.write( "    * To make it easier, multiply by 100 to remove decimals: `4x + 6y ≤ 1200000`<BR>\n" );
document.write( "    * Divide by 2 to simplify: `2x + 3y ≤ 600000`<BR>\n" );
document.write( "    * Graph the line `2x + 3y = 600000`.<BR>\n" );
document.write( "    * Find the intercepts:<BR>\n" );
document.write( "        * If x = 0, then 3y = 600000, so y = 200000. (0, 200000)<BR>\n" );
document.write( "        * If y = 0, then 2x = 600000, so x = 300000. (300000, 0)<BR>\n" );
document.write( "    * Draw the line connecting these points.<BR>\n" );
document.write( "    * Since it's `2x + 3y ≤ 600000`, we shade the region below the line.\r<BR>\n" );
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document.write( "3.  **x ≥ 0 and y ≥ 0:**<BR>\n" );
document.write( "    * This means we are only considering the first quadrant of the coordinate plane.\r<BR>\n" );
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document.write( "To fully graph this, plot the points and shade the appropriate regions. The solution region is where all shaded areas overlap within the first quadrant.\r<BR>\n" );
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document.write( "**c. Determining the Amount of Savings**\r<BR>\n" );
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document.write( "1.  **Graphing and Intersection:**<BR>\n" );
document.write( "    * The solution to the system of inequalities is the region where the graphs of all inequalities overlap.<BR>\n" );
document.write( "    * The corner points of this region are important. They represent possible solutions.<BR>\n" );
document.write( "    * The exact amounts of savings can be determined by finding the coordinates of the intersection point of the lines `x + y = 150000` and `2x + 3y = 600000`.\r<BR>\n" );
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document.write( "2.  **Solving the System of Equations:**<BR>\n" );
document.write( "    * Solve the system of equations:<BR>\n" );
document.write( "        * `x + y = 150000`<BR>\n" );
document.write( "        * `2x + 3y = 600000`<BR>\n" );
document.write( "    * From the first equation, `x = 150000 - y`.<BR>\n" );
document.write( "    * Substitute this into the second equation: `2(150000 - y) + 3y = 600000`<BR>\n" );
document.write( "    * `300000 - 2y + 3y = 600000`<BR>\n" );
document.write( "    * `y = 300000`<BR>\n" );
document.write( "    * Substitute `y = 300000` back into `x = 150000 - y`: `x = 150000 - 300000 = -150000`<BR>\n" );
document.write( "    * Because x is negative, this corner point is outside of the feasible region. Therefore the intersection of the two lines is outside of the solution set. We must find another method.<BR>\n" );
document.write( "    * Test corner points that are created by the intersection of the individual lines, with the x or y axis.<BR>\n" );
document.write( "    * Test the point (150000,0).<BR>\n" );
document.write( "        * 150000+0 = 150000. 0.04(150000) + 0.06(0) = 6000. This point is valid.<BR>\n" );
document.write( "    * Test the point (0,150000).<BR>\n" );
document.write( "        * 0+150000 = 150000. 0.04(0) + 0.06(150000) = 9000. This point is valid.<BR>\n" );
document.write( "    * Test the point (300000,0).<BR>\n" );
document.write( "        * 300000+0 = 300000. 0.04(300000) + 0.06(0) = 12000. This point is valid.<BR>\n" );
document.write( "    * Test the point (0,200000).<BR>\n" );
document.write( "        * 0+200000 = 200000. 0.04(0)+0.06(200000) = 12000. This point is valid.\r<BR>\n" );
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document.write( "**d. Possible Amounts of Savings**\r<BR>\n" );
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document.write( "Here's one possible solution:\r<BR>\n" );
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document.write( "* Mrs. Diez could have Php 150,000 in the 4% bank (x = 150000) and Php 0 in the 6% bank (y = 0).<BR>\n" );
document.write( "    * Total savings: 150000 + 0 = 150000<BR>\n" );
document.write( "    * Total interest: (0.04 * 150000) + (0.06 * 0) = 6000\r<BR>\n" );
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document.write( "Another possible solution.<BR>\n" );
document.write( "* Mrs. Diez could have Php 0 in the 4% bank (x = 0) and Php 150,000 in the 6% bank (y = 150000).<BR>\n" );
document.write( "    * Total savings: 0 + 150000 = 150000<BR>\n" );
document.write( "    * Total interest: (0.04 * 0) + (0.06 * 150000) = 9000.\r<BR>\n" );
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document.write( "Another possible solution.<BR>\n" );
document.write( "* Mrs. Diez could have Php 300,000 in the 4% bank (x=300000) and Php 0 in the 6% bank (y=0).<BR>\n" );
document.write( "    * Total savings: 300000+0=300000<BR>\n" );
document.write( "    * Total interest: (0.04 * 300000) + (0.06 * 0) = 12000.\r<BR>\n" );
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document.write( "Another possible solution.<BR>\n" );
document.write( "* Mrs. Diez could have Php 0 in the 4% bank (x=0) and Php 200000 in the 6% bank (y=200000).<BR>\n" );
document.write( "    * Total savings: 0+200000=200000<BR>\n" );
document.write( "    * Total interest: (0.04 * 0) + (0.06 * 200000) = 12000.<BR>\n" );
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