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Using Matrices to Solve Systems of Equations

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Matrix Equations l We have solved systems using graphing, but now we learn how to do it using matrices. This will be particularly useful when we have equations with three variables.

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Matrix Equation l Before you start, make sure that both of your equations are in standard form and the variables are in the same order (alphabetical usually is best).

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Setting up the Matrix Equation l Given a system of equations -2x - 6y = 0 3x + 11y = 4 l Since there are 2 equations, there will be 2 rows. l Since there are 2 variables, there will be 2 columns.

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l There are 3 parts to a matrix equation 1)The coefficient matrix, 2)the variable matrix, and 3)the constant matrix. Setting up the Matrix Equation

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-2x - 6y = 0 3x + 11y = 4 l The coefficients are placed into the coefficient matrix.

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-2x - 6y = 0 3x + 11y = 4 l Your variable matrix will consist of a column.

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-2x - 6y = 0 3x + 11y = 4 l The matrices are multiplied and represent the left side of our matrix equation.

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-2x - 6y = 0 3x + 11y = 4 l The right side consists of our constants. Two equations = two rows.

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-2x - 6y = 0 3x + 11y = 4 l Now put them together. We’ll solve it later!

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Create a matrix equation l 3x - 2y = 7 y + 4x = 8 l Put them in Standard Form. l Write your equation.

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3a - 5b + 2c = 9 4a + 7b + c = 3 2a - c = 12 Create a matrix equation

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l To solve matrix equations, get the variable matrix alone on one side. l Get rid of the coefficient matrix by multiplying by its inverse Solving a matrix equation

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l When solving matrix equations we will always multiply by the inverse matrix on the left of the coefficient and constant matrix. (remember commutative property does not hold!!)

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l The left side of the equation simplifies to the identity times the variable matrix. Giving us just the variable matrix.

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l Using the calculator we can simplify the left side. The coefficient matrix will be A and the constant matrix will be B. We then find A -1 B.

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l The right side simplifies to give us our answer. l x = -6 l y = 2 l You can check the systems by graphing, substitution or elimination.

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Advantages l Basically, all you have to do is put in the coefficient matrix as A and the constant matrix as B. Then find A -1 B. This will always work!!! l NO SOLVING FOR Y!!!!! :)

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Solve: l Plug in the coeff. matrix as A l Put in the const. matrix as B l Calculate A -1 B.

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Solve: l r - s + 3t = -8 l 2s - t = 15 l 3r + 2t = -7

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Word Problem Systems l The sum of three numbers is 12. The 1st is 5 times the 2nd. The sum of the 1st and 3rd is 9. Find the numbers.

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Word Problem Systems l The sum of three numbers is 12. l x + y + z = 12 l The 1st is 5 times the 2nd. l x = 5y l The sum of the 1st and 3rd is 9. l x + z = 9

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Word Problem Systems l x + y + z = 12 x = 5y => x - 5y = 0 x + z = 9

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Word Problem Systems l x + y + z = 12 x - 5y = 0 x + z = 9

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