A system of linear equations includes two or more equations of two or more variables, which are all raised to the power of 1. In order to solve a system of linear equations, you must have at least one equation for each unknown variable. Matrices allow you to solve systems of equations more quickly, particularly when there are more than two unknowns.
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Write each equation in the form of Axe + By + Cz = D, where A, B, and C are the coefficients for variables x, y, and z, and D is a constant. Your equations may have fewer or greater variables, depending on how many terms they contain.
Create a coefficient matrix, a variable matrix, and a constant matrix. The coefficient matrix will represent the coefficients of each of your equations. For example, if your equations were A1x + B1y = C1 and A2X + B2Y = C2, your coefficient matrix would be [A1 B1; A2 B2], where a semicolon represents the start of a new row. Your variable matrix would be [x; y], and your constant matrix would consist of the numbers on the right-hand side of your equations, or [C1; C2].
Write the system of equations in matrix form. If you let the coefficient matrix = [C], the variable matrix = [V], and the constant matrix equal [S], you can model the system of equations as [C]*[V] = [S].
Solve the matrix equation for the variable matrix, [V]. You can do this by multiplying each side by the inverse coefficient matrix: [C]^-1[C][V] = [C]^-1[S]. The [C]^-1 and [C] cancel out to yield [V] = [C]^-1[S].
Define the coefficient matrix in the TI-84 calculator. Hit "2nd" then "MATRX", and then use the side arrows to get to the "MATRX EDIT" menu. Type in the number of rows in the coefficient matrix, hit "ENTER," type in the number of columns, and then hit "ENTER" again. Enter the value of each coefficient in the matrix when the rectangular cursor prompts you for each element. You need to press "ENTER" after every value.
Enter the constant matrix, [S], according to the same process you used to enter the coefficient matrix. Make sure you enter [S] in a new matrix space rather than overwriting [C].
Press "2nd," and then "QUIT" to return to the home screen.
Enter the expression, [C]^-1*[S] to solve for [V]. Hit "2nd," and then "MATRX" for the matrix menu, and select the matrix for [C]. Hit the inverse key (which looks like "x^-1") to find [C]^-1. Then multiply [C]^-1 by [S] to get the value for your variable matrix, [V].
Tips and warnings
- Note that [C]^-1 represents the inverse matrix of [C] and not 1/[C].
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