4-8 Cramer’s Rule We can solve a system of linear equations that has a unique solution by using determinants and a pattern called Cramer’s Rule (named.

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4-8 Cramer’s Rule We can solve a system of linear equations that has a unique solution by using determinants and a pattern called Cramer’s Rule (named after its creator Gabriel Cramer)

Cramer’s Rule uses the ratio of two determinants to solve for each variable The denominator of the ratio is the determinant of the coefficients of the x and y values in the system of equations The numerator of the ratio to solve for x is arrived at by replacing the first column of the coefficient matrix with the constants of the answer in the system of equations The numerator of the ratio to solve for y is arrived at by replacing the second column of the coefficient matrix with the constants of the answer in the system of equations

To use Cramer’s Rule to solve for x and y in the following system: 3x + 5y = 19 4x + 3y = 7 The denominator used in all of the ratios in the problems is the determinant of the coefficients of each of the variables 3(3) – 5(4) = =

The numerator used to find the value of x in the problem is found by replacing the first column of the matrix (the x column) with the constants (the answers) and leaving the second column alone. Find the determinant of that matrix (19)(3) – (5)(7) = 57 –35 = 22

The numerator used to find the value of y in the problem is found by replacing the second column of the matrix (the y column) with the constants (the answers) and leaving the first column alone. Find the determinant of that matrix (3)(7) – (19)(4) = 21 – 76 = -55

The value of x is found by finding the ratio of the determinant of x over the determinant of the original matrix and reducing the ratio x = = -2

The value of y is found by finding the ratio of the determinant of y over the determinant of the original matrix and reducing the ratio y = = 5

The answers are (-2, 5) Check your answers in both equations 3x + 5y = 19 4x + 3y = 7 3(-2) + 5(5) = 194(-2) + 3(5) = = = 7 19 = 19 7 = 7