\n" );
document.write( "
\n" );
document.write( " Lets start with the given system of linear equations
\n" );
document.write( "
\n" );
document.write( " 
\n" );
document.write( " 
\n" );
document.write( "
\n" );
document.write( " Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.
\n" );
document.write( "
\n" );
document.write( " Solve for y for the first equation
\n" );
document.write( "
\n" );
document.write( "  Subtract  from both sides
\n" );
document.write( "
\n" );
document.write( "  Divide both sides by -3.
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " Which breaks down and reduces to
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Now we've fully isolated y
\n" );
document.write( "
\n" );
document.write( " Since y equals  we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x.
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Replace y with . Since this eliminates y, we can now solve for x.
\n" );
document.write( "
\n" );
document.write( "  Distribute 1 to
\n" );
document.write( "
\n" );
document.write( "  Multiply
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " Reduce any fractions
\n" );
document.write( "
\n" );
document.write( "  Subtract  from both sides
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Make 1 into a fraction with a denominator of 3
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Combine the terms on the right side
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Make 3 into a fraction with a denominator of 3
\n" );
document.write( "
\n" );
document.write( "  Now combine the terms on the left side.
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Multiply both sides by  . This will cancel out  and isolate x
\n" );
document.write( "
\n" );
document.write( " So when we multiply  and (and simplify) we get
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  <---------------------------------One answer
\n" );
document.write( "
\n" );
document.write( " Now that we know that , lets substitute that in for x to solve for y
\n" );
document.write( "
\n" );
document.write( "  Plug in into the 2nd equation
\n" );
document.write( "
\n" );
document.write( "  Multiply
\n" );
document.write( "
\n" );
document.write( "  Add  to both sides
\n" );
document.write( "
\n" );
document.write( "  Combine the terms on the right side
\n" );
document.write( "
\n" );
document.write( "  Multiply both sides by  . This will cancel out 1 on the left side.
\n" );
document.write( "
\n" );
document.write( "  Multiply the terms on the right side
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Reduce
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " So this is the other answer
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " <---------------------------------Other answer
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " So our solution is
\n" );
document.write( "
\n" );
document.write( " and
\n" );
document.write( "
\n" );
document.write( " which can also look like
\n" );
document.write( "
\n" );
document.write( " ( , )
\n" );
document.write( "
\n" );
document.write( " Notice if we graph the equations (if you need help with graphing, check out this solver)
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " we get
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  graph of (red) and (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " and we can see that the two equations intersect at (,). This verifies our answer.
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " -----------------------------------------------------------------------------------------------
\n" );
document.write( " Check:
\n" );
document.write( "
\n" );
document.write( " Plug in (,) into the system of equations
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " Let and . Now plug those values into the equation
\n" );
document.write( "
\n" );
document.write( "  Plug in and
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Multiply
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Add
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " Reduce. Since this equation is true the solution works.
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " So the solution (,) satisfies
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " Let and . Now plug those values into the equation
\n" );
document.write( "
\n" );
document.write( "  Plug in and
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Multiply
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "  Add
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " Reduce. Since this equation is true the solution works.
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " So the solution (,) satisfies
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " Since the solution (,) satisfies the system of equations
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " this verifies our answer.
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( " |
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
