Which represents the solution of the system of equations, y=x^2-x+1 and y=x? determine the solution set by graphing

(1,1)

Step-by-step explanation:

We have the following system:

We are going to plug the 2nd equation into the 1st eqaution giving us:

Now time to solve .

Subtract x on both sides:

Now this actually not too bad.  This a perfect square trinomial.  That is it is of the form which equals .

So solving is equivalent to solving

In order to solve  , we just need to know when x-1=0 which is at x=1.  I got by adding 1 on both sides of x-1=0

Now remember y=x means we have the solution (1,1).

Okay, I'm going to leave all of this algebra here.  I'm going to graph this without a graphing calculator.

is a line with slope 1 and y-intercept 0.

is a parabola.  The vertex isn't obvious to me right now without the algebra but I do know the parabola is open up because the coefficient of x^2 being positive.

So let's find the vertex.

I'm going to start with the x-coordinate which is -b/(2a).

Compare to and you should see that .

So

So the vertex occurs when the x-coordinate is 1/2.

To find the correspond y-value just use the equation that relations x and y.

That is use .

Replace x with 1/2.

Find a common denominator.

So the vertex is at (1/2,3/4) and the parabola is open up.

When you plug in 0 into you get so the ordered pair (0,1) is on the parabola.

Using symmetry about the line x=1/2 we know that (1,1) is also on this graph.

Let's use this information to produce a graph now.

Now our answer appears to be (1,1) in the graph.

We can check this answer by plugging in (1,1) into our system and see if both equations check out:

 gives us 1=1 which is true.  So we are good there.

gives us which is true. So we are good there.

The work for showing .

1

So the point (1,1) checks out for both of our equations which means it is a common point amongst the equations given.

whole numbers.

step-by-step explanation:

we are given the average monthly rainfall (in millimeters) model r(m) for the years 1900-2009 as a function of the month number m.

as we can see m is taken for the number of months for years 1900-2009.

we know that number of months can't be a negative number or decimal number or a fraction.

so, the number of months could be only positive whole numbers.

therefore, whole numbers is more appropriate for the domain of r.