Answer:
Explanation:
To solve the system of equations graphically, we'll plot both equations on the same set of axes and find the points where the two graphs intersect. These points represent the solutions to the system. The system of equations is:
1. \(y = x^2 + 2x + 6\)
2. \(y = 2x + 6\)
Let's graph these two equations:
1. \(y = x^2 + 2x + 6\)
This equation represents a parabola. To graph it, you can find a few key points or use the vertex form to identify the vertex. The vertex form of a parabola is \(y = a(x - h)^2 + k\), where (h, k) is the vertex.
In this case, the vertex can be found using the formula \(h = -b/(2a)\), where \(a = 1\) and \(b = 2\) from the equation \(y = x^2 + 2x + 6\):
\(h = -2/(2*1) = -1\)
Now, substitute \(h = -1\) into the equation to find \(k\):
\(k = (-1)^2 + 2(-1) + 6 = 1 - 2 + 6 = 5\)
So, the vertex of the parabola is (-1, 5), and it opens upward.
2. \(y = 2x + 6\)
This equation represents a straight line with a slope of 2 and a y-intercept of 6.
Now, let's plot both equations on the same graph:
- The parabolic equation \(y = x^2 + 2x + 6\) has its vertex at (-1, 5) and opens upward.
- The linear equation \(y = 2x + 6\) has a slope of 2 and a y-intercept of 6.
When you graph these two equations, you'll notice that they intersect at one point. This point represents the solution to the system of equations.
The solution to the system is the point where the two graphs intersect, which is approximately (-2, 2). So, the solution is x = -2 and y = 2.