document.write( "Question 948496: When graphing a system of linear inequalities how do you determine what the solution of the system is? \n" ); document.write( "

Algebra.Com's Answer #578961 by MathLover1(20849) $\"\"$ $\"About$

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When we talk about the $\"solution\"$ of the system of equations, we mean the $\"values\"$ of the variables that make both equations $\"true\"$ at the $\"same\"$ time.
\n" ); document.write( "There may be $\"many\"$ pairs of $\"x\"$ and $\"y\"$ that make the $\"first\"$ equation $\"true\"$ , and $\"many\"$ pairs of $\"x\"$ and $\"y\"$ that make the $\"second\"$ equation $\"true\"$ , but we are looking for an $\"x\"$ and $\"y\"$ that would work in $\"BOTH\"$ equations.\r
\n" ); document.write( "\n" ); document.write( "A system of two linear equations in two unknowns might look like:\r
\n" ); document.write( "\n" ); document.write( " $\"2x%2B4y=3\"$
\n" ); document.write( " $\"x-3y=1\"$ \r
\n" ); document.write( "\n" ); document.write( "see their graph:\r
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-2x%2F4%2B3%2F4%2C+x%2F3-1%2F3%29+\"$ \r
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\n" ); document.write( "\n" ); document.write( "If equations are linear, their graphs will be straight lines that intersect at $\"one\"$ particular $\"point\"$ $\"if\"$ the system $\"have\"$ a solution. Clearly this point is on $\"both\"$ lines, and therefore its coordinates ( $\"x\"$ , $\"y\"$ ) will satisfy the equation of either line. Thus the pair ( $\"x\"$ , $\"y\"$ ) is the $\"ONE\"$ and $\"ONLY\"$ solution to the system of equations.\r
\n" ); document.write( "\n" ); document.write( "Sometimes two equations might look different but actually describe the same line. For example, in\r
\n" ); document.write( "\n" ); document.write( " $\"2x%2B3y=1\"$
\n" ); document.write( " $\"4x%2B6y=2\"$ \r
\n" ); document.write( "\n" ); document.write( "The second equation is just two times the first equation, so they are actually $\"equivalent\"$ and would both be equations of the $\"same\"$ line. Because the two equations describe the same line, they have all their points in common; hence there are an $\"INFINITE\"$ number of solutions to the system.\r
\n" ); document.write( "\n" ); document.write( "see their graph:\r
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-2x%2F3%2B1%2F3%2C+-4x%2F6%2B2%2F6%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "If two lines happen to have the $\"same\"$ slope, but are $\"not\"$ identically the $\"same\"$ line, then they will $\"NEVER\"$ intersect. There is $\"NO\"$ pair ( $\"x\"$ , $\"y\"$ ) that could $\"satisfy\"$ $\"+both\"$ equations, because there is no point ( $\"x\"$ , $\"y\"$ ) that is simultaneously on both lines. Thus these equations are said to be $\"INCONSISTENT\"$ , and there is $\"NO\"$ solution. The fact that they both have the $\"same\"$ $\"+slope\"$ may not be obvious from the equations, because they are not written in one of the standard forms for straight lines. The slope is not readily evident in the form we use for writing systems of equations. (If you think about it you will see that the $\"slope\"$ is the negative of the coefficient of $\"+x\"$ divided by the coefficient of $\"y\"$ ).\r
\n" ); document.write( "\n" ); document.write( "example:\r
\n" ); document.write( "\n" ); document.write( " $\"y=2x%2B3\"$
\n" ); document.write( " $\"y=2x-1\"$ \r
\n" ); document.write( "\n" ); document.write( "see their graph:\r
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+2x%2B3%2C+2x-1%29+\"$ \r
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