Solve Using an Augmented Matrix 7x-7y=-7 , -3x+3y=3

Problem

{[7*x−7*y=−7],[−3*x+3*y=3])

Solution

1. Write the augmented matrix for the system of linear equations by extracting the coefficients and the constants.

[[7,−7,|,−7],[−3,3,|,3]]

2. Simplify the first row by dividing all elements in (R_1) by 7 to obtain a leading 1

[[1,−1,|,−1],[−3,3,|,3]]

3. Eliminate the x-term in the second row by performing the row operation (R_2)←(R_2)+3*(R_1)

[[1,−1,|,−1],[0,0,|,0]]

4. Interpret the result of the row-echelon form. Since the second row consists entirely of zeros (0=0, the system is dependent and has infinitely many solutions.

5. Express the solution in terms of a parameter. Let y=t where t is any real number. From the first row, x−y=−1 which means x=y−1

x=t−1

Final Answer

{[x=t−1],[y=t]),for any real number *t

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