Question 86048
Lets start with the given system of linear equations

{{{8*x-4*y=16}}}
{{{y=2x-4}}}



Since y equals {{{2x-4}}} we can substitute the expression {{{2x-4}}} into y of the 1st equation. This will eliminate y so we can solve for x.


{{{8*x+-4*highlight((2x-4))=16}}} Replace y with {{{2x-4}}}. Since this eliminates y, we can now solve for x.

{{{8*x-4*(2x )-4(-4 )x=16}}} Distribute -4 to {{{2x-4}}}


{{{8*x-8x+ 16=16}}} Multiply


{{{8*x-8*x=16-16 }}} Subtract {{{16}}} from both sides


{{{8*x-8*x=0 }}} Combine the terms on the right side



{{{0*x=0 }}} Now combine the terms on the left side.  
 {{{0 =0 }}} Since this expression is true for any x, we have an identity.


So there are an infinite number solutions. The simple reason is the 2 equations represent 2 lines that overlap each other. So they intersect each other at an infinite number of points.

If we graph {{{-2x+y=-4}}} and {{{y=2x-4}}} we get

{{{ graph( 500, 600, -6, 5, -10, 10, (-4--2*x)/1) }}} graph of {{{-2*x+1*y=-4}}}


we can see that these two lines are the same. So this system is <a href=http://www.algebra.com/algebra/homework/coordinate/lessons/Types-of-systems-inconsistent-dependent-independent.lesson>dependent</a>



*[invoke solving_linear_system_by_substitution 4, -12, 5, -1, 3, -1]