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The solutions to a system of equations are the points of intersection of the lines. For a system with two quadratic equations, there are 4 cases to consider: 2 solutions, 1 solution, no solutions, and infinite solutions.

Basic concepts: System of linear-quadratic equations, Solving quadratic equations by factoring, Solving quadratic equations using the quadratic formula,

Related concepts: Graphing linear inequalities in two variables, Graphing systems of linear inequalities, Graphing quadratic inequalities in two variables, Graphing systems of quadratic inequalities,

- 1.• The solutions to a system of equations are the points of intersection of the graphs.

• For a system consisting of two quadratic equations:

quadratic equation: $y = a{x^2} + bx + c$

quadratic equation: $y = d{x^2} + ex + f$

There are 4 cases to consider:case 1: 2 solutions case 2: 1 solution case 3: no solutions case 4: infinite solutions

- 2.
**Case 1: System with 2 Solutions**

Solve the system, then verify the solutions graphically:

$y = {x^2} - 6x + 5$

$y = - 2{x^2} + 9x - 7$

- 3.
**Case 2: System with 1 Solution**

Solve the system, then verify the solutions graphically:

$y = 2{x^2} + 6x + 7$

$y = - {x^2} + 4$

- 4.
**Case 3: System with No Solutions**

Solve the system, then verify the solutions graphically:

$y = - {x^2} + 6x - 10$

$y = 2{x^2} + 6x + 5$

- 5.
**Case 4: System with Infinite Solutions**

Solve the system, then verify the solutions graphically:

${x^2} - 4x - y + 3 = 0$

$5y - 5{x^2} + 20x - 15 = 0$

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