Solve using augmented matrices. y = 9 -10x - 6y = -14

Steps to Solve

1. Convert equations to standard form Both equations need to be in the standard form ( Ax + By = C ).

   From the first equation, ( y = 9 ) can be rewritten as:

   $$ 0x + 1y = 9 $$

   The second equation, ( -10x - 6y = -14 ), remains as is.

2. Set up the augmented matrix Using the coefficients from the equations, we can write the augmented matrix. The coefficients of ( x ), ( y ), and the constants give us:

   $$ \begin{pmatrix} 0 & 1 & | & 9 \ -10 & -6 & | & -14 \end{pmatrix} $$

3. Apply row operations to solve We want to manipulate the augmented matrix to reach row echelon form.

   Start by swapping the two rows to place the second equation first:

   $$ \begin{pmatrix} -10 & -6 & | & -14 \ 0 & 1 & | & 9 \end{pmatrix} $$

   Next, we can simplify the first row. Divide the first row by -10:

   $$ \begin{pmatrix} 1 & 0.6 & | & 1.4 \ 0 & 1 & | & 9 \end{pmatrix} $$

4. Back substitution to find the solution From the second row ( y = 9 ), substitute ( y ) back into the first row equation:

   $$ x + 0.6(9) = 1.4 $$

   Calculate:

   $$ x + 5.4 = 1.4 \implies x = 1.4 - 5.4 = -4 $$

5. Final solution Thus, the solution to the system is ( x = -4 ) and ( y = 9 ).