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Solving Systems of Equations by Substitution

Solving a system by graphing is certainly limited by the accuracy of the graph. If the lines intersect at a point whose coordinates are not integers, then it is difficult to determine those coordinates from the graph. The method of solving a system by substitution does not depend on a graph and is totally accurate. For substitution we replace a variable in one equation with an equivalent expression obtained from the other equation. Our intention in this substitution step is to eliminate a variable and to give us an equation involving only one variable.


Example 1

An independent system solved by substitution

Solve the system by substitution:

2x + 3y = 8

y + 2x = 6


We can easily solve y + 2x = 6 for y to get y = -2x + 6. Now replace y in the first equation by -2x + 6:

2x + 3y = 8  
2x + 3(-2x + 6) = 8 Substitute -2x + 6 for y.
2x - 6x + 18 = 8  
-4x = -10  

To find y, we let in the equation y = -2x + 6:

The next step is to check and y = 1 in each equation. If and y = 1 in 2x + 3y = 8, we get

If and y = 1 in y + 2x = 6, we get

Because both of these equations are true, the solution set to the system is . The equations of this system are independent.


Example 2

An inconsistent system solved by substitution

Solve by substitution:

x - 2y = 3

2x - 4y = 7


Solve the first equation for x to get x - 2y = 3 . Substitute 2y + 3 for x in the second equation:

2x - 4y = 7
2(2y + 3) = 7
4y + 6 - 4y = 7
6 = 7

Because 6 = 7 is incorrect no matter what values are chosen for x and y, there is no solution to this system of equations. The equations are inconsistent. To check, we write each equation in slope-intercept form:

x - 2y = 3 2x - 4y = 7
-2y = -x + 3 -4y = -2x + 7
y y

The graphs of these equations are parallel lines with different y-intercepts. The solution set to the system is the empty set, Ø.


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