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Solving Systems of Linear Equations by Substitution

Solving Systems of Linear Equations by Substitution

Solving Systems of Linear Equations by Substitution

Solving Systems of Linear Equations by Substitution

Solving Systems of Linear Equations by Substitution

Solving Systems of Linear Equations by Substitution

Solving Systems of Linear Equations by Substitution

Graphing is a useful tool for solving systems of equations, but it can sometimes be time-consuming. A quicker way to solve systems is to isolate one variable in one equation, and substitute the resulting expression for that variable in the other equation. Observe:
Example 1: Solve the following system, using substitution:


5x + y = 13  
3x = 15 - 3y  

The easiest variable to isolate is y in the first equation, because it has no coefficient:

y = 13 - 5x
In the second equation, substitute for y its equivalent expression:
3x = 15 - 3(13 - 5x)
Solve the equation:
3x = 15 - 39 + 15x
3x = 15x - 24
-12x = - 24
x = 2
Now substitute this x -value into the "isolation equation" to find y :
y = 13 - 5x = 13 - 5(2) = 13 - 10 = 3
Thus, the solution to the system is (2, 3) . It is useful to check this solution in both equations.

Note: Although we chose y in the first equation in the previous example, isolating any variable in any equation will yield the same solution.

Example 2: Solve the following system, using substitution:


2x + 4y = 36  
10y - 5x = 0  

It is easier to work with the second equation, because there is no constant term:

5x = 10y
x = 2y
In the first equation, substitute for x its equivalent expression:
2(2y) + 4y = 36
Solve the equation:
4y + 4y = 36
8y = 36
y = 4.5
Plug this y -value into the isolation equation to find x :
x = 2y = 2(4.5) = 9
Thus, the solution to the system is (9, 4.5).

Example 3: Solve the following system, using substitution:


2x - 4y = 12  
3x = 21 + 6y  

It is easiest to isolate x in the second equation, since the x term already stands alone:

x =
x = 7 + 2y
In the first equation, substitute for x its equivalent expression:
2(7 + 2y) - 4y = 12
Solve the equation:
14 + 4y - 4y = 12
14 = 12

Since 14≠12 , the system of equations has no solution. It is inconsistent (and independent). The two equations describe two parallel lines.

Example 4: Solve the following system, using substitution:


10x = 4y - 68  
2y - 5x = 34  

Either equation can be used to isolate the variable. We will isolate y in the second equation:

2y = 5x + 34
y =
y = x + 17
In the first equation, substitute for y its equivalent expression:
10x = 4( x + 17) - 68
10x = 10x + 68 - 68
10x = 10x
0 = 0

Since 0 = 0 for any value of x , the system of equations has infinite solutions. Every ordered pair ( x, y ) which satisfies y = x + 17 (the isolation equation) is a solution to the system. The system is dependent (and consistent). The two equations describe the same line-- y = x + 17 .