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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Systems of Equations That Have No Solution or Infinitely Many Solutions

What happens when a system of equations has no solution or infinitely many solutions?

This question is best addressed by examples.

Example 1

Solve the system of equations.

2x + 6y = 9

x + 3y = 4

Solution

Solve the second equation for x to get x = 4 - 3y. Then substitute 4 - 3y for x in the first equation.

 2x + 6y = 9 2(4- 3y ) + 6y = 9 Substitute 4 - 3y for x. 8 - 6y + 6y = 9 Distributive Property 8 9

Since this equation is never true, there are no solutions to the system. The graph of this system of equations is shown below. Notice that the lines are parallel. Key Idea

When solving a system of equations, if the final equation does not contain a variable and is false, then there are no solutions.

Example 2

Solve the system of equations.

2x + 8y = 14

x + 4y = 7

Solution

Solve the second equation for x to get x = 7 - 4y. Then substitute 7 - 4y for x in the first equation.

 2x + 8y = 14 2(7 - 4y ) +8y = 14 Substitute 7 - 4y for x. 14 - 8y + 8y = 14 14 = 14

This equation does not involve either variable, but the equation is always true. In this case, there are infinitely many solutions. The two lines are identical.

Key Idea

When solving a system of equations, if the final equation does not contain a variable and is true, then there are infinitely many solutions.