Question

Explain how you know the following system of equations has at most 2 solutions.

Solve the system algebraically. Show your work.
Solve the system by graphing.
y = 2x - 1
y = x² - 9
a) The system has at most 2 solutions because one equation is linear and the other is a quadratic equation.
b) The system has at most 2 solutions because both equations are linear.
c) The system has at most 2 solutions because both equations are quadratic.
d) The system has at most 2 solutions because one equation is linear and the other is a cubic equation.

Answer 1

The system has at most 2 solutions because one equation is linear and the other is a quadratic equation. The algebraic solution involves substituting the linear equation into the quadratic equation and solving for the variables. The graphical solution involves plotting the equations on a coordinate plane and finding the intersection points.

Step-by-step explanation:

The system has at most 2 solutions because one equation is linear and the other is a quadratic equation. To solve the system algebraically, we can substitute the expression for y from the linear equation into the quadratic equation. This gives us:

x^2 - 9 = 2x - 1

By rearranging the equation:

x^2 - 2x = 8

Then:

x^2 - 2x - 8 = 0

Using the quadratic formula, we find that x = -2 or x = 4. Substituting these values back into the linear equation, we can find the corresponding y-values.

To solve the system by graphing, we can plot the graphs of the two equations on the same coordinate plane. The points where the graphs intersect represent the solutions to the system. By analyzing the graph, we can see that there are at most 2 intersection points, confirming our algebraic solution.