Question 176056
Not quite correct.


A consistent system has <i><b>at least 1</i></b> solution.


An inconsistent system has no solutions.


A dependent system has an infinite number of solutions.


An independent system is uniquely determined, that is there is <i><b>exactly 1</i></b> solution.



{{{y=abs(x-2)+3}}} means:


{{{y = x - 2 + 3}}} &#8594; {{{y = x + 1}}} for all {{{x >= 2}}} and {{{y = - (x - 2) + 3}}} &#8594; {{{y = -x + 5}}} for all {{{x <= 2}}},


So:  {{{y = x + 1}}} for {{{x >= 2}}}.  Substitute in {{{y=(x-2)^2+3}}} &#8594; {{{x + 1 = (x - 2)^2 + 3}}} &#8594; {{{x + 1 = x^2 - 4x + 4 + 3}}} &#8594; {{{x^2 - 5x + 6 = 0}}} &#8594; {{{(x - 3)(x - 2) = 0}}} &#8594; {{{x = 3}}} or {{{x = 2}}}


{{{x = 2}}} &#8594; {{{y = 3}}} and {{{x = 3}}} &#8594; {{{y = 4}}}


Also:  {{{y = -x + 5}}} for {{{x <= 2}}}.  Substitute in {{{y=(x-2)^2+3}}} &#8594; {{{-x + 5 = (x - 2)^2 + 3}}} &#8594; {{{-x + 5 = x^2 - 4x + 4 + 3}}} &#8594; {{{x^2 - 3x + 2 = 0}}} &#8594; {{{(x - 2)(x - 1) = 0}}} &#8594; {{{x = 2}}} or {{{x = 1}}}.


{{{x = 2}}} &#8594; {{{y = 3}}} and {{{x = 1}}} &#8594; {{{y = 4}}}


Hence there are three ordered pairs that satisfy the system:


*[Tex \LARGE \left(2,3\right)] and 


*[Tex \LARGE \left(3,4\right)] and


*[Tex \LARGE \left(1,4\right)]


Now let's go back and review the definitions:


A consistent system has <i><b>at least 1</i></b> solution.  <i>Yes, this system is consistent because it has at least 1 solution</i>


An inconsistent system has no solutions. <i>No, this system is not inconsistent because it has at least 1 solution</i>


A dependent system has an infinite number of solutions. <i>No, this system is not dependent because it has only three solutions, not an infinite number of solutions.  A dependent system also has the characteristic that any ordered pair that satisfies one equation must also satisfy the other.  In this case (0,5) would satisfy the absolute value equation, but not the quadratic.</i>


An independent system is uniquely determined, that is, there is <i><b>exactly 1</i></b> solution. <i>No, this system is not independent because there is more than one solution.</i>


See graphs of the two functions illustrating the three points of intersection:


{{{drawing(
600,600,-1,9,-1,9,
grid(1),
circle(2,3,.05),
locate(2,2.9,P1(2,3)),
circle(3,4,.05),
locate(3,3.8,P2(3,4)),
circle(1,4,.05),
locate(1,4.4,P3(1,4)),
graph(600,600,-1,9,-1,9,
(x - 2)^2 + 3,
abs(x - 2) + 3
))
}}}