Question

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.
(3/8)x - (1/2)y = 7/8
-6x + 8y = -14

Accepted Answer

Write the system of equations in matrix form as an augmented matrix. For the system: $$\frac{3}{8}x - \frac{1}{2}y = \frac{7}{8}$$ and $$-6x + 8y = -14$$, the augmented matrix is:

$$\left[\begin{array}{cc|c} \frac{3}{8} & -\frac{1}{2} & \frac{7}{8} \ -6 & 8 & -14 \end{array}ight]$$ Use row operations to create a leading 1 in the first row, first column. You can multiply the first row by the reciprocal of $$\frac{3}{8}$$, which is $$\frac{8}{3}$$, to make the coefficient of $$x$$ equal to 1. Next, eliminate the $$x-$$term in the second row by adding a suitable multiple of the first row to the second row. Specifically, multiply the first row by 6 (to match the $$-6 in $$the second row) and add it to the second row to make the $$x$$ coefficient in the second row zero. Then, create a leading 1 in the second row, second column (the $$y$$ coefficient) by dividing the entire second row by the coefficient of $$y in $$that row after elimination. Finally, eliminate the $$y-$$term in the first row by adding or subtracting a suitable multiple of the second row from the first row to get zeros above and below the leading 1s, resulting in the reduced row echelon form. From this form, write the solution for $$x$$ and $$y.$$

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Key Concepts

Gauss-Jordan Elimination Method

Types of Solutions for Linear Systems

Expressing Solutions with Arbitrary Variables