At your school book fair, paperbacks cost one price and hardcovers cost another. You buy 3 paperbacks and 2 hardcovers. Your total comes to $54. Your best friend buys 2 paperbacks and 4 hardcovers. His total comes to $76. How could you use a matrix to find the price of each type of book?
Previously we learned how to systems of linear equations using graphing, substitution and linear combinations. In this topic, we will explore one way to use matrices and determinants to solve linear systems.
Cramer’s Rule in Two Variables:
Cramer’s Rule in Three Variables:
So, using the formulas above:
Therefore, the solution is (2, -1).
Because the determinant is zero, there is no unique solution and we cannot solve the system further using Cramer's Rule. Looking at the system, we see that the left-hand side of the first equation is a multiple of the second equation, by 3. The right sides are not multiples of each other, therefore there is no solution.
So the solution is (2, -3, 5).
Use Cramer’s Rule to solve the systems below.
2. Find the
Therefore, there is no unique solution. We must use linear combination or the substitution method to determine whether there are an infinite number of solutions or no solutions. Using linear combinations we can multiply the first equation by 2 and get the following:
3. Find the
Therefore, the solution is (-6, 1, 4).
Solve the systems below using Cramer’s Rule. If there is no unique solution, use an alternate method to determine whether the system has infinite solutions or no solution.
Solve the systems below using Cramer’s Rule. You may wish to use your calculator to evaluate the determinants.