PrerequisitesPrerequisite

Solving three-variable systems

Eliminate one variable to form a two-variable system, then work back.

  • Use two pairs of equations to eliminate the same variable.
  • Solve the resulting two-variable system.
  • Substitute into an original equation to find the third variable.

Example

Eliminate first

Solve the system of equations for all three variables.

Equation (2): subtract times equation (1) to eliminate .Equation (3): subtract times equation (1) to eliminate .This is equation (2) after dividing every term by .Equation (1): subtract times equation (2) to eliminate .
Equation (3): subtract times equation (2) to eliminate .This is equation (3) after dividing every term by .Equation (1): subtract times equation (3) to eliminate .Read each variable and check all values in the original equations.

Your turn

Question 1

Solve the system.

Check answer
Equation (2): subtract times equation (1) to eliminate .Equation (3): subtract times equation (1) to eliminate .This is equation (2) after dividing every term by .Equation (1): subtract times equation (2) to eliminate .
Equation (3): subtract times equation (2) to eliminate .This is equation (3) after dividing every term by .Equation (1): subtract times equation (3) to eliminate .Read each variable and check all values in the original equations.
Question 2

Solve the system.

Check answer
Equation (2): subtract times equation (1) to eliminate .Equation (3): subtract times equation (1) to eliminate .This is equation (2) after dividing every term by .Equation (1): subtract times equation (2) to eliminate .
Equation (3): subtract times equation (2) to eliminate .This is equation (3) after dividing every term by .Equation (1): subtract times equation (3) to eliminate .Read each variable and check all values in the original equations.
Question 3

Solve the system.

Check answer
Equation (2): subtract times equation (1) to eliminate .Equation (3): subtract times equation (1) to eliminate .This is equation (2) after dividing every term by .Equation (1): subtract times equation (2) to eliminate .
Equation (3): subtract times equation (2) to eliminate .This is equation (3) after dividing every term by .Equation (1): subtract times equation (3) to eliminate .Read each variable and check all values in the original equations.
Question 4

Solve the system.

Check answer
Equation (2): subtract times equation (1) to eliminate .Equation (3): subtract times equation (1) to eliminate .This is equation (2) after dividing every term by .Equation (1): subtract times equation (2) to eliminate .
Equation (3): subtract times equation (2) to eliminate .This is equation (3) after dividing every term by .Equation (1): subtract times equation (3) to eliminate .Read each variable and check all values in the original equations.
Question 5

Solve the system.

Check answer
Equation (2): subtract times equation (1) to eliminate .Equation (3): subtract times equation (1) to eliminate .This is equation (2) after dividing every term by .Equation (1): subtract times equation (2) to eliminate .
Equation (3): subtract times equation (2) to eliminate .This is equation (3) after dividing every term by .Equation (1): subtract times equation (3) to eliminate .Read each variable and check all values in the original equations.
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