Relationship between Equations in Linear Systems. Systems of Equations with One Linear and One Quadratic Equation - Revision Notes

In these revision notes for Relationship between Equations in Linear Systems. Systems of Equations with One Linear and One Quadratic Equation, we cover the following key points:

• What are the examples of special cases of linear equation systems?
• What are independent systems of linear equations? How many solutions do they have?
• What are independent systems dependent and inconsistent systems of linear equations. How many solutions do they have?
• How many points do the graphs of equations involved in a dependent system of linear equations have?
• How many points do the graphs of equations involved in a dependent and inconsistent systems of linear equations have?
• How do we solve systems of equations where one equation is linear and one quadratic?
• How to solve systems of equations where both of them are quadratic? How many possible solutions do such systems have?

Relationship between Equations in Linear Systems. Systems of Equations with One Linear and One Quadratic Equation Revision Notes

The systems of linear equations that give a specific number pair as solutions are called independent systems. The graphs of the linear equations in such a system have a single point in common, where the coordinates of this point represent the solutions of the system.

The systems of equations that have no solution are called inconsistent. The graphs that represent each equation in such systems are parallel, i.e. they have no point in common.

If in a system of linear equations there is an infinity of possible solutions, where all solutions of the first equation are also true for the second one, it is obvious that we are dealing with equations that have the same graph. Such systems of equations are called dependent, as well as the equations contained in them.

If, in a system of equations, one equation is linear and the other is quadratic we have the following possible relations between the variables and equations' graphs based on the number of points they have in common. Thus,

1. When the two equations have two points in common (two pairs of solutions), like the system in the above example, the graphs are said to be intersecting.
2. When the two graphs have a single point in common (one pair of solutions), the graphs are said to be tangent.
3. When the two graphs have no points in common (no solutions), the graphs are said to be divergent. The system of equations in this case is inconsistent.

If both equations in a system are quadratic, we may have from zero to three pairs of numbers as a solution set. The only difference in the procedure of solutions with the systems where one of the equations was linear and the other quadratic consists of the fact that now we have to substitute the square of a given variable instead of a variable in the first power. Then, the rest of the procedure is the same.