11.6 Solving systems with gaussian elimination  (Page 4/13)

Write the augmented matrix.

First, multiply row 1 by $\mathrm{-1}$ to get a 1 in row 1, column 1. Then, perform row operations    to obtain row-echelon form.

The last matrix represents the following system.

We see by the identity $0=0$ that this is a dependent system with an infinite number of solutions. We then find the generic solution. By solving the second equation for $y$ and substituting it into the first equation we can solve for $z$ in terms of $x.$

Now we substitute the expression for $z$ into the second equation to solve for $y$ in terms of $x.$

The generic solution is $\left(x,\frac{2\mathrm{-2}x}{3},\frac{1-x}{3}\right).$

Write the augmented matrix for the system of equations.

On the matrix page of the calculator, enter the augmented matrix above as the matrix variable $\left[A\right].$

Use the ref( function in the calculator, calling up the matrix variable $\left[A\right].$

Evaluate.

Using back-substitution, the solution is $\left(\frac{61}{187},-\frac{92}{187},-\frac{24}{187}\right).$

We have a system of two equations in two variables. Let $x=$ the amount invested at 10.5% interest, and $y=$ the amount invested at 12% interest.

As a matrix, we have

Multiply row 1 by $\mathrm{-0.105}$ and add the result to row 2.

Then,

So $\mathrm{12,000}\mathrm{-5,000}=\mathrm{7,000.}$

Thus, $5,000 was invested at 12% interest and $7,000 at 10.5% interest.

We have a system of three equations in three variables. Let $x$ be the amount invested at 5% interest, let $y$ be the amount invested at 8% interest, and let $z$ be the amount invested at 9% interest. Thus,

As a matrix, we have

Now, we perform Gaussian elimination to achieve row-echelon form.

The third row tells us $-\frac{1}{3}z=\mathrm{-2,000};$ thus $z=\mathrm{6,000.}$

The second row tells us $y+\frac{4}{3}z=\mathrm{9,000.}$ Substituting $z=\mathrm{6,000},$ we get

The first row tells us $x+y+z=10,000.$ Substituting $y=1,000$ and $z=6,000,$ we get

The answer is $3,000 invested at 5% interest, $1,000 invested at 8%, and $6,000 invested at 9% interest.