Solution (Systems of Equations)

Definition

A solution to a system of equations is a set of values for the variables that satisfy all the equations in the system simultaneously. In other words, the solution is the point (or points) where the graphs of the equations intersect, if the system has a solution.

Key Points:

• A solution represents the values of the variables that make all the equations in the system true.

• The solution can be expressed as an ordered pair (x, y) for a system of two linear equations, where x is the value for the first variable and y is the value for the second variable.

Types of Solutions in a System of Equations:

1. One solution (Unique solution): The system of equations has exactly one set of values for the variables. Graphically, this occurs when the lines (or planes) intersect at exactly one point.

2. No solution: The system has no solution if the equations represent parallel lines (or planes) that never intersect. The lines are inconsistent and do not share any common point.

3. Infinite solutions: The system has infinitely many solutions if the equations represent the same line (or plane). The lines coincide and overlap completely, meaning every point on the line is a solution.

Example:

Consider the system of equations:

1. y = 2x + 1

2. y = -x + 4

To find the solution, set the two equations equal to each other:

2x + 1 = -x + 4

Solve for x:

2x + x = 4 - 1 → 3x = 3 → x = 1

Now, substitute x=1 into either original equation to find y:

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

So, the solution is (1, 3).

This is the unique solution to the system because the two lines intersect at (1, 3).

Graphically:

When you graph both equations, the point (1, 3) is where the two lines meet, and this is the solution to the system of equations.