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Solving Systems of Equations by Substitution


A dependent system solved by substitution

Solve by substitution:

2x + 3y = 5 + x + 4y
y = x - 5


Substitute y = x - 5 into the first equation:

2x + 3(x - 5) = 5 + x + 4(x - 5)
2x + 3x - 15 = 5 + x + 4x - 20
5x - 15 = 5x - 15

Because the last equation is an identity, any ordered pair that satisfies y = x - 5 will also satisfy 2x + 3y = 5 + x + 4y. The equations of this system are dependent. The solution set to the system is the set of all points that satisfy y = x -  5. We write the solution set in set notation as

{(x, y) | y = x - 5}.

We can verify this result by writing 2x + 3y = 5 + x + 4y in slope-intercept form:

2x + 3y = 5 + x + 4y
3y = -x + 5 + 4y
-y = -x + 5
y = x - 5

Because this slope-intercept form is identical to the slope-intercept form of the other equation, they are two equations that look different for the same straight line.

Heplful Hint

The purpose of this Example is to show what happens when a dependent system is solved by substitution. If we had first written the first equation in slope-intercept form, we would have known that the equations are dependent and would not have done substitution.

If a system is dependent, then an identity will result after the substitution. If the system is inconsistent, then an inconsistent equation will result after the substitution. The strategy for solving an independent system by substitution can be summarized as follows.


The Substitution Method

1. Solve one of the equations for one variable in terms of the other.

2. Substitute into the other equation to get an equation in one variable.

3. Solve for the remaining variable (if possible).

4. Insert the value just found into one of the original equations to find the value of the other variable.

5. Check the two values in both equations.


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