Solve A Systems of Linear Equations by Substitution

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Substitution Instructions Problems w/Solutions
Substitution Means Expression Swapping

    Substitution means expression swapping. One expression is substituted or used in place of another expression of identical value so that a variable is eliminated.

    Though this method always works, it is often easier to use another method unless one equations is already solved for one variable.

    IF ONE EQUATION IS ALREADY SOLVED FOR ONE VARIABLE, use substitution.

    Compare the two systems.

5x+4y=6
y=-2x+3
  
2x+y=3
5x+4y=6

    It's really the same system. The terms and format is just ordered differently.

    The light blue one, on the left, is ideal for solution by substitution since one equations is already solved for one variable.

    The light green one, on the right, is ideally set up for linear combination, or determinants.

    Below the light blue one is solved by substitution.

    So, to find the values of x and y, add lines together in a certain way.

    Here's the instructions.

Use Substitution to Solve Systems of Equations
1st: Pick 1 equation and solve for 1 of the variables.
2nd: Substitute this expression in the other equation.
3rd: Solve.
4th: Use this new value in the step 1 equation to get the other variable.
5th: State the solution and include values for both variables.

    Here's an example.

Solve a system by Substitution
1st: Pick 1 equation and solve for 1 of the variables.
 
5x+4y=6
y=
-2x+3
 
The second equation is solved for y.
 
"-2x+3" is the expression needed.
 
2nd: Substitute this expression in the other equation.
5x+4y=6
5x+4( )=6
5x+4(-2x+3 )=6
 
3rd: Solve.
5x+4(-2x+3)=6
5x-8x+12=6
-3x+12=6
-12-12
-3x=-6
-3x/(-3)=-6/(-3)
x=2
 
4th: Use this new value in the step 1 equation to get the other variable.
y=-2x+3
y=-2(x)+3
y=-2( )+3
y=-2(2)+3
y=-4+3
y=-1
 
5th: State the solution and include values for both variables.
x = 2,   y = -1
(2, -1)


Pick the systems best suited to solution by substitution.
  If you picked correctly, "yes" is the message when you mouseover the 1st arrow.  
  Solve the best suited systems for practice.  
  A mouseover the 2nd arrow yields the solution to the system.  
x - 2y = -2
-3x - 6y = -6 no dependent system, coincident lines, many solutions, (0,1) for example		 y=4x - 2
x=3 yes (3,10)		 5x + 5y = 10
10x + 10y = 12 no no solution, parallel lines, inconsistent system		 4x - 3y=12
x=3y-1yes (13/3,16/9
4x+10y=-22
y=3x-9 yes (2,-3)		 y = -2x + 4
y = -2x + 5 yes no solution, parallel lines, inconsistent system		 y=4x+2
2y=8x+4 yes no solution, parallel lines, inconsistent system		 y = 4x - 1
8y = 8x + 8 yes no solution, parallel lines, inconsistent system
-2x - 2y = -4
3x - 8y = -12 no (3/5,7/5)		 x+3y=6
3x+9y=18 no no solution, parallel lines, inconsistent system		 4x + 4y = 10
x + y = 12 no no solution, parallel lines, inconsistent system		 y=x+2
y=4x+2 yes (0,2)
2x +2y = 6
y=-x+3yes dependent system, coincident lines, many solutions, (0,3) for example		 -3x+3y=6
6x+9y=12 no (-2/5,8/5)		 3x+ 3=y
9x-3y=-9 yes dependent system, coincident lines, many solutions, (0,3) for example		 5x+15y=10
3x+9y=6 no dependent system, coincident lines, many solutions, (0,2/3) for example


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