SOLUTION: Solve the system by either substitution or the elimination method, if possible. (If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, enter INFINITEL
Question 911401: Solve the system by either substitution or the elimination method, if possible. (If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, enter INFINITELY MANY.)
3x + 12y = −45
x = 3y + 27
This requires a little explanation but is remarkably powerful. You can represent any system of equations as two matrices, linear transformations between vector spaces, one containing the coefficients of the equations and the other the solutions. This is usually written as AX = B, where A is the matrix of coefficients, X is the desired n-dimensional vector, and B the matrix of constant terms. To solve this equation we can use Cramer's formula stating: x_i = det(A_i)/det(A) where i = 1,..,n, where det( ) is the determinant of which there is a fantastic explanation here: http://mathinsight.org/determinant_linear_transformation
and A_i is obtained from A by substituting the ith column of B.
Now we can use the definition of the determinant to get the formulas used in the above solver. If, on the other hand, the above solver doesn't make sense you can proceed by substitution as follows: take the second equation and plug it into the first: 3(3y + 27) +12y = -45
Now simplify to 9y+81+12y = -45, which further simplifies to: 21y+81 = -45 or 18y = -126 or y = -6. Now sub this into the second equation: 3(-6)+27 = 9. For a quick and efficient way to check # of solutions see:http://www.algebra.com/algebra/homework/coordinate/Linear-systems.faq.question.911749.html