System of Linear Equations

Systems with no solution and infinite solutions

These cases only apply to systems of two lines. If you are working with systems with three or more linear equations (lines), you cannot use the blanket generalizations made below.

• The system has exactly 1 solution.
  • Systems have 1 and only 1 solution when the two lines have different slope. Think about it, if the two lines have different slopes then eventually at some point they must meet. After all the lines are not parallel.


• system has no solutions
  • Systems have no solution when the lines are parallel (ie have the same slope) and the lines have different y-intercepts.
    • As an example look at the following two lines
    • Line 1: y = 5x +13
    • Line 2: y = 5x + 12


• The system has infinite solutions
  • Systems have infinite solutions when the lines are parallel and the lines have the same y-intercept. If two lines have the same slope (ie are parallel) and the same y-intercept, they are actually the same exact line. In other words, systems have infinite solutions when the two lines are the same line!
    • As an example consider the following two lines
      • Line 1: y = x +3
      • Line 2: 2y = 2x +6
      These two lines are exactly the same line. If you multiply line 1 by two you get line 2.

• graph : by looking at where lines intersect (meet) on a graph
• algebraic equation : by setting the equations of the system equal to each other then solving this equation.
• substitution : by solving for one of the variables and substituting its value in to the other equation.
• Elimination : Elimination involves algebraic manipulations of two or more equations. The end goal is to eliminate a variable by creating opposite coefficients (The examples below should clarify this straightforward approach).

• y=x+1
• y=2x

Example Problem

• y = 2x+1
• y = 4x - 1

• 2y = 4x + 2
and
• 2y = -x + 7

• Line 1: y=3x – 1
• Line 2: y= x – 5

Set the Two Equations Equal to each other then solve for x

 step 2  
1) Substitute the x value, -2, into either equation to determine y coordinate of solution
2) y=(-2) - 5 -7
3) The solution is the point (-2, -7)

• Line 1: y= x + 1
• Line 2: y= 2x

The solution of this system is the point of intersection : (1,2).

Below is the graph showing this solution.

• Line 1: y= x +5
• Line 2: y= 2x +2

The solution of this system is the point of intersection : (3,8).

Below is the graph showing this solution.

• Line 1: y= x – 1
• Line 2: y= 2x +2

The solution of this system is the point of intersection : (-3,-4).

• Line 1: y= x + 2
• Line 2: y= x + 8

This system of linear equation has no solution

Substitution Example Two
• Line 1 : y=2x+1
•          Line 2 : 2y=3x-2
• Step 1: Substitute one equation into the other
  • 2(2x+1)=3x-2
• Step 2: Now that you have a single variable equation, solve for that variable's equation
  •  4x+2 = 3x-2
  • x+2= -2
  • x= –4
• Step 3 : Once you have solved for the one variable insert that variable back into either equation to obtain the value of y at the solution.
  • Insert x= –4 to find y value y = 2(–4)+1= –7
  This example's solution is ( –4, –7).

Show Graph

• Line 1: y= x + 1
• Line 2: y= 2x

The solution of this system is (2,6).

• Line 1: y = 5x – 1
• Line 2: y= 3x + 12

This system of lines has a solution at the point (2,9).

• Line 1: y = 3x + 1
• Line 2: 4y = 12x + 4

This system has an infinite number of solutions.

• Line 1: y = 3x + 1
• Line 2: 4y = 12x + 3

Whenever you arrive at a contradiction such as 3 = 4, your system of linear equations has no solutions. When you use these methods (substitution, graphing etc...) to find the solution what you're really asking is at what point (which x coordinate and y coordinate) do the two lines intersect. However, if the only time that they intersect is when 3 = 4, they are never going to intersect since 3 does not equal four!

This system has an no solutions.

Elimination Method

Example of Elimination

• Line 1: y = x + 1
• Line 2: y = –x