document.write( "Question 1091985: (x+1)^2-(y-1)^2=20, x^2-(y+2)^2= 24 \n" ); document.write( "

Algebra.Com's Answer #706509 by ikleyn(52817) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "You are given the system of two equations in two unknowns; each equation is of degree 2:\r\n" );
document.write( "\r\n" );
document.write( "(x+1)^2-(y-1)^2 = 20,          (1)\r\n" );
document.write( "x^2    -(y+2)^2 = 24.          (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Your first step is to expand:\r\n" );
document.write( "\r\n" );
document.write( "x^2 + 2x + 1 - y^2 + 2y - 1 = 20,\r\n" );
document.write( "x^2          - y^2 - 4y - 4 = 24.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Simplify:\r\n" );
document.write( "\r\n" );
document.write( "x^2 + 2x - y^2 + 2y = 20,      (3)\r\n" );
document.write( "x^2      - y^2 - 4y = 28.      (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Subtract (4) from (3) (both sides):\r\n" );
document.write( "\r\n" );
document.write( "     2x + 6y        =  -8,     (5)\r\n" );
document.write( "x^2      - y^2 - 4y =  28.     (6)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now you have one equation of degree 1 (linear equation (5)) and one equation of degree 2.\r\n" );
document.write( "\r\n" );
document.write( "So, you reduced the degree,  and it was your first major step.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now, multiply eq(6) by 4 (both sides)\r\n" );
document.write( "\r\n" );
document.write( "     2x + 6y        =   -8,    (7)\r\n" );
document.write( "4x^2    - 4y^2 - 16y = 112.    (8)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From (7), express  2x = -8- 6y;  ====>  4x^2 = (-8 - 6y)^2 = (8 + 6y)^2 and substitute this expression for 4x^2 into (8). \r\n" );
document.write( "You will get\r\n" );
document.write( "\r\n" );
document.write( "(8 + 6y)^2 - 4y^2 - 16y = 112   (9)    instead of (8).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus you reduced the system to the single quadratic equation for y, and it was our second major step.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now it is simple arithmetic to solve it.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "When you get  y,  calculate  x. \r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Are these instructions enough for you ?\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "You can send your response or your questions to me through the \"Thank you\" form.
\n" ); document.write( "In this case do not forget to refer to the problem ID number 1091985.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To see more solved problems (your samples) of this type, see the lesson\r
\n" ); document.write( "\n" ); document.write( " - Solving the system of algebraic equations of degree 2 \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this online textbook under the topic \"Systems of equations that are not linear\".\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------
\n" ); document.write( "Lesson to learn from this solution.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "    First major step is to reduce the given system of two eqs of degree 2 to \r\n" );
document.write( "    one equation of degree 2 and one eqn of degree 1 (linear eqn).\r\n" );
document.write( "\r\n" );
document.write( "    It is \"the degree reducing step\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Second major step is reducing the simplified system to a single quadratic equation.\r\n" );
document.write( "    It is achieved using substitution from the linear equation.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    When these major steps are completed, the rest is arithmetic.\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );