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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:

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

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

4.

Systems of Equations

4.1

Determining number of solutions to linear equations

4.2

Solving systems of linear equations by graphing

4.3

Using elimination method to solve systems of equations

4.4

Using substitution method to solve systems of equations

4.5

Money related questions in linear equations

4.6

Unknown number related questions in linear equations

4.7

Distance and time related questions in linear equations

4.8

Rectangular shape related questions in linear equations

4.9

System of quadratic-quadratic equations

4.10

Solving 3 variable systems of equations by substitution

4.11

Solving 3 variable systems of equations by elimination

4.12

Solving 3 variable systems of equations (no solution, infinite solutions)

4.13

Word problems relating 3 variable systems of equations

We have over 1100 practice questions in Algebra 2 for you to master.

Get Started Now4.1

Determining number of solutions to linear equations

4.3

Using elimination method to solve systems of equations

4.4

Using substitution method to solve systems of equations

4.5

Money related questions in linear equations

4.6

Unknown number related questions in linear equations

4.7

Distance and time related questions in linear equations

4.8

Rectangular shape related questions in linear equations

4.9

System of quadratic-quadratic equations