Solution of a System of Linear Equations, how to solve systems using substitution, the equations of the lines themselves and graphs.

Systems with no solution and infinite solutions

When you are trying to calculate the solution of a system of linear equations, you can will arrive at one of three distinct cases:

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

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.

The solution of the system of equations on the left is (2,2) which marks the point where the two lines intersect.

To find the solution to systems of linear equations, you can any of the methods below:

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).
Use a graphing calculator to find the intersections of the graphs. Yeah, a calculator can do pretty much all of the work for you.
Use this system of equations solver and just enter the linear equations then hit 'solve'!

Interactive System of Linear Equations

The Graph Method

On the left, the system of linear equations is the following two lines:

y=x+1
y=2x

What is the solution?
answer

Example Problem

Soltuion to System Of Linear Equations, Graph

Use the graph method to solve the system of equations on the left.

y = 2x+1
y = 4x - 1

answer

Solve the following system of linear equations by graphing.

2y = 4x + 2
2y = -x + 7

Answer

The Algebraic Equation Method

Let's take another look at the system of equations from above:

y=2x+1
y=4x-1

By examining the graph we can see that the point of intersection, or the solution, is the point (1,3) where the lines intersected.

Steps for the algebraic method:

make sure that each linear equation is reduced to slope intercept form
- (ie y=3x+2 is good but 2y=6x+4 is NOT)
set the two equations equal to each other
- 2x+1=4x-1
Solve for X
- 2x+1=4x-1
- 2=2x
- x= 1
insert x value into either equation to determine y coordinate of solution

4(1)-1=3

The solution is the ordered pair you've just calculated
- (1,3)

Practice Problem

Practice Problem Five

What is the solution to the following system of linear equations:

Line 1: y=3x – 1
Line 2: y= x – 5

Step 1

Practice Problem Six

Set the two equations equal to calculate the solution to the system below:

Line 1: y= x + 1
Line 2: y= 2x

Answer

Practice Problem Seven

Solve the system of linear equations by setting their equations equal:

Line 1: y= x +5
Line 2: y= 2x +2

Answer

Practice Problem Eight

Solve the system of linear equations by setting their equations equal:

Line 1: y= x – 1
Line 2: y= 2x +2

Answer

Practice Problem Nine

Solve the system of linear equations by setting their equations equal:

Line 1: y= x + 2
Line 2: y= x + 8

Answer

The Substitution Method

The substitution method involves algebraic substitution of one equation into a variable of the other.

A quick refresher on algebraic substitution:

Refresher: Substitution

Equation 1 : x = 5
Equation 2: y = x +2

1) Use equation 1( x= 5) to substitute 5 for x in second equation
- y = (5) + 2
2) So love for Y

y = 5+ 2 = 7

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).

Let's examine the graph of this system to see if we correctly solved the problem.

Show Graph

Practice Problems

Practice Problem One

Use the substitution method to solve the system:

Line 1: y= x + 1
Line 2: 2y= 3x

Answer

Practice Problem Two

Use the substitution method to solve the system:

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

Answer

Practice Problem Three

Use substitution to solve the following system of linear equations:

Line 1: 2x – 3y = 6
Line 2: x + y = -12

Practice Problem Four

Use substitution to solve the system:

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

Answer

Practice Problem Five

Use substitution to solve the system:

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

Answer

Elimination Method

Elimination method is an algebraic method for solving systems. To use elimination you perform an operation on 1 equation then add the two equations so that one of the variables cancels.

Line 1: y = x + 1
Line 2: y = –x

System of Linear Equations

How to Solve Systems

Systems with no solution and infinite solutions

Example Problem

Elimination Method