12x+a=131 what is the value of a if x=7
You asked:
MathBot Answer:
The system of linear equations has exactly one solution.
\[a = 47\]
Begin by writing the augmented matrix of the system of equations. $$\begin{bmatrix}1 & 12 &\bigm |& 131\\0 & 1 &\bigm |& 7\end{bmatrix}$$
Then use a series of elementary row operations to convert the matrix into reduced-row echelon form. The three elementary row operations are:
1. Swap the positions of any two rows.
2. Multiply any row by a nonzero scalar.
3. Multiply a row by a nonzero scalar and add it to any other row.
First, switch the rows in the matrix such that the row with the leftmost non-zero entry with the greatest magnitude is at the top.
$$\begin{bmatrix}1 & 12 &\bigm |& 131\\0 & 1 &\bigm |& 7\end{bmatrix}$$
The leading term of row \(1\) is already \(1\) so this row does not need to be multiplied by a scalar.
$$\begin{bmatrix}1 & 12 &\bigm |& 131\\0 & 1 &\bigm |& 7\end{bmatrix}$$
The leading term of row \(2\) is already \(1\) so this row does not need to be multiplied by a scalar.
$$\begin{bmatrix}1 & 12 &\bigm |& 131\\0 & 1 &\bigm |& 7\end{bmatrix}$$
Multiply row \(2\) by scalar \(-12\) and add it to row \(1\).
$$\begin{bmatrix}1 & 0 &\bigm |& 47\\0 & 1 &\bigm |& 7\end{bmatrix}$$
Once the matrix is in reduced-row echelon form, convert the matrix back into linear equations to find the solution. $$\begin{aligned}1 \cdot a+ 0 \cdot x = 47 \\ a = 47\end{aligned}$$$$\begin{aligned}0 \cdot a+ 1 \cdot x = 7 \\ x = 7\end{aligned}$$