We want to solve the given system of equations.
To solve the system of equations means
to determine the ordered pairs in the form of x comma y
that satisfy both equations.
We are given y equals three x plus two
and y equals negative x squared plus two.
Because both equations are solve for y,
we will solve this system using the method of substitution
where because y is equal to three x plus two,
we can substitute three x plus two for y
in the second equation.
By doing this, we will then have an equation
and one variable that we can then solve for x.
But then we'll have to come back
and determine the corresponding y values.
Remember, each solution is an ordered pair
in the form of x comma y.
So performing the substitution,
we have the equation three x plus two
equals negative x squared plus two.
Notice how we have a degree two equation
or a quadratic equation,
so let's set the equation equal to zero
and see if we can solve by factoring.
We normally prefer to have the x squared term be positive,
and therefore we will set the right side equal to zero
by adding x squared to both sides,
as well as subtracting two.
And now simplifying on the left,
we have x squared plus three x.
And then two minus two is zero.
So on the left we have x squared plus three x equals
on the right side negative x squared plus x squared is zero
and two minus two is zero,
which is the reason why we added x squared to both sides,
as well as subtracted two on both sides.
So the right side is now equal to zero.
And now let's factor the left side.
Remember, the first step in factoring is
to factor out the greatest common factor.
Greatest common factor of x squared plus three x is x.
We factor out x.
We're left with the quantity x plus three.
So now we have x times the quantity
x plus three equals zero.
The product on the left is equal to zero when x equals zero
or the factor of x plus three is equal to zero.
So one solution to the equation is x equals zero.
And now if we solve this equation for x,
we subtract three on both sides
and we have x equals negative three.
But we're not done.
Remember, a solution consists of an x and a y value
or an ordered pair in the form of x comma y.
So we found two possible x values.
We now have to find the corresponding y values.
So our solutions are going to be two ordered pairs
where one ordered pair, the x value is negative three.
And for the second ordered pair, the x value is zero.
So now we need to use one of the original equations
to determine the corresponding y values.
Let's go ahead and use a linear equation,
y equals three x plus two.
So when x is equal to negative three,
y must be equal to three times negative three plus two.
Three times negative three is negative nine.
Negative nine plus two is negative seven.
So one solution is the ordered pair
negative three comma negative seven.
And now we need to determine the corresponding y value
when x is zero.
So when x is zero,
we have y equals three times zero plus two,
which is equal to zero plus two, which is two.
So another solution is the ordered pair zero comma two.
Again, remember, each ordered pair is
in the form of x comma y.
So the system of equations has two solutions.
And let's verify this graphically.
To verify it graphically,
we graph both equations on the coordinate plane,
and these two ordered pairs
should represent the points of intersection.
So in blue, we have the graph of y equals three x plus two.
In red, we have the graph
of y equals negative x squared plus two.
Notice how we have two point of intersection
where one ordered pair
is negative three comma negative seven
and the other ordered pair is zero comma two,
which verifies our solutions are correct.
I hope you found this helpful.