It is often desirable or even necessary to use more than one variable to model situations in many fields. We write and solve a system of equations in order to answer questions about the situation.

If a system of linear equations has at least one solution, it is **
consistent**. If the system has no solutions, it is **inconsistent**. If
the system has an infinity number of solutions, it is **dependent**.
Otherwise it is **independent**.

A linear equation in three variables is an equation equivalent to the
equation

where A, B, C, and D are real numbers and A, B, C, and D are not all 0. This is the equation of a plane.

Work the following problems. Click on *Solution*, if you want to review the solutions.**
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A total of $50,000 is invested in three funds paying 6%, 8%, and 10%
simple interest. The yearly interest is $3,700. Twice as much money is
invested at 6% as invested at 10%. How much is invested in each of the funds.

**Problem 3.1b:**

The standard equation of a circle is
*x*^{2}+*y*^{2}+*Ax*+*By*+*C*=0. Find the
equation of the circle that passes through the points
,
,
and

**Problem 3.1c:**

Your company has three acid solutions on hand: 30%, 40%, and 80% acid. It
can mix all three to come up with a 100-gallons of a 39% acid solution. If
it interchanges the amount of 30% solution with the amount of the 80%
solution in the first mix, it can create a 100-gallon solution that is 59%
acid. How much of the 30%, 40%, and 80% solutions did the company mix
to create a 100-gallons of a 39% acid solution?

**Problem 3.1d:**

Five hundred tickets were sold for a certain music concert. The tickets for
the adults sold for $7.50, the tickets for the children sold for $4.00,
and tickets for senior citizen sold for $3.50. The revenue for the Monday
performance was $3,025. Twice as many adult tickets were sold as children tickets. How many
of each ticket was sold?

**Problem 3.1e:**

Solve the following system of equations for x, y and z:

**Problem 3.1f:**

Solve the following system of equations for x, y and z:

If you would like to return to the beginning of the two by two system of equations, click on
**start**.

If you would like to review three-variable systems example, click on
**three**.

**
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**Author**: Nancy Marcus

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