# Solve \begin{cases} 2x +5y = 15\\4x +y =21 \end{cases} a) Write down its Augmented Matrix b)...

## Question:

Solve {eq}\begin{cases} 2x +5y = 15\\4x +y =21 \end{cases} {/eq}

a) Write down its Augmented Matrix,

b) Reduce the system to reduced row-echelon form by doing the equivalent operations on the rows,

c) Write down the solution of the system if it is consistent.

## Solving Systems of Equations with Matrices

If we have multiple equations relating multiple variables, we can solve the system by constructing an augmented matrix and performing row operations. This may lead us to a unique solution, or it may tell us that the solution has infinitely many solutions or no solutions at all.

a) An augmented matrix can be constructed from this linear system. We can take the coefficients on each variable as the elements in this matrix.

{eq}\begin{bmatrix} 2 & 5 & 15\\ 4 & 1 & 21 \end{bmatrix} {/eq}

b) Let's now conduct row operations to solve this system. We can add multiples of rows together, swap rows, and multiply rows by constants.

{eq}\begin{align*} \begin{bmatrix} 2 & 5 & 15\\ 4 & 1 & 21 \end{bmatrix} & R_2 = R_2 - 2R_1\\ \begin{bmatrix} 2 & 5 & 15\\ 0 & -9 & -9 \end{bmatrix} & R_2 = -\frac{1}{9}R_2\\ \begin{bmatrix} 2 & 5 & 15\\ 0 & 1& 1 \end{bmatrix} &R_1 = R_2 - 5R_2\\ \begin{bmatrix} 2 & 0 & 10\\ 0 & 1& 1 \end{bmatrix} &R_1 = \frac{1}{2}R_1\\ \begin{bmatrix} 1& 0 & 5\\ 0 & 1& 1 \end{bmatrix} \end{align*} {/eq}

c) This is a consistent system, as we were able to find a single solution by performing row operations. The solution set is that {eq}x = 5 {/eq} and {eq}y = 1 {/eq}.