A system of equations can have two solutions if it is a non-linear system, specifically if it involves quadratic equations. But for linear system of equation it is not possible to have two solutions. They either can have one, infinite or no solution.

What is a System of Equations?

A system of equations is a set of two or more equations with the same set of unknowns. The solution to the system is the set of values for the unknowns that satisfy all equations simultaneously.

Linear systems of equations can be classified into three types based on the number of solutions they possess:

1. Consistent and Independent: One unique solution.
2. Consistent and Dependent: Infinitely many solutions.
3. Inconsistent: No solution.

Example of System of Equation with Two Solution

A common example is when you have a system involving a quadratic equation and a linear equation.

For instance, consider the system:

• y = x² + 1
• y = 2x + 3

To find the solutions, we can set the equations equal to each other:

x² + 1 = 2x + 3

Rearranging gives us a quadratic equation:

x² – 2x – 2 = 0

Solving this quadratic equation using the quadratic formula [Tex] x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}[/Tex]:

[Tex]x = \frac{2 \pm \sqrt{4 + 8}}{2} = \frac{2 \pm \sqrt{12}}{2} = \frac{2 \pm 2\sqrt{3}}{2} = 1 \pm \sqrt{3}[/Tex]

So, we have two solutions for x:

[Tex]x = 1 + \sqrt{3} \quad \text{and} \quad x = 1 – \sqrt{3}[/Tex]

Substituting these values back into one of the original equations to find y:

y = 2x + 3

For x = 1 + √3:

y = 2(1 + √3) + 3 = 2 + 2√3 + 3 = 5 + 2√3

For x = 1 – √3:

y = 2(1 – √3) + 3 = 2 – 2√3 + 3 = 5 – 2√3

Thus, the system has two solutions:

(1 + √3, 5 + 2√3) and (1 – √3, 5 – 2√3)

Geometric Interpretation of this Example

Another way to visualize this is by considering the intersection points of a parabola and a line on a graph. A line can intersect a parabola at two distinct points, leading to two solutions for the system.

Read More,

System of Linear Equations

Linear Equations

Linear Equations in One Variable

Pair Of Linear Equations In Two Variables

L U Decomposition of a System of Linear Equations

FAQs

What is a system of equations?

A system of equations is a set of two or more equations that have common variables and are solved simultaneously.

Can linear systems have exactly two solutions?

No, linear systems cannot have exactly two solutions. They either have one solution, no solution, or infinitely many solutions.

What types of equations can form a system with exactly two solutions?

Nonlinear equations, such as quadratic equations or combinations of linear and quadratic equations, can form systems with exactly two solutions.