Solving Systems of Linear Equations

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In this activity, you will discover formulas for the solutions of $2\times 2$ linear systems.

Consider the following $2\times 2$ linear system of equations. Our goal is to derive an expression for $x$ in terms of the letters $a,b,c,d,p,$ and $q$ .

$\begin{eqnarray} ax+by&=&p\\ cx+dy&=&q \end{eqnarray}$

Multiply Equation (1) by $d$ : $\answer {adx+bdy=pd}$

Multiply Equation (2) by $-b$ : $\answer {-bcx-bdy=-bq}$

Add the two new equations from Problems 1 and 2:

$\answer {adx-bcx+ bdy-bdy =pd-bq}$

You should have two terms whose sum is $0$ in order to eliminate the $y$ variable.

Solve for $x$ in your equation from Problem 3.

$x=\answer {\frac {pd-bq}{ad-bc}}$

$adx-bcx=(ad-bc)x$

Next, we will solve for $y$ in the same $2\times 2$ linear system given by Equations (1) and (2).

Multiply Equation (1) by $-c$ : $\answer {-acx-bcy=-pc}$

Multiply Equation (2) by $a$ : $\answer {acx+ady=qa}$

Add the two new equations from Problems 5 and 6:

$\answer {-acx+acx -bcy+ady=-pc+qa}$

There should be two terms that cancel out to eliminate the $x$ variable.

Solve for $y$ in your equation from Problem 7:

$y=\answer {\frac {-pc+qa}{-bc+ad}}$

$-bcy+ady=(-bc+ad)y$

Lastly, we will solve for $x$ in the following $3\times 3$ linear system of equations

$\begin{eqnarray} ex+fy+gz&=&r\\ hx+iy+jz&=&s\\ kx+ly+mz&=&t \end{eqnarray}$

by doing the following:

Multiply Equation (3) by $-j$ : $\answer {-jex-jfy-jgz=-jr}$

Multiply Equation (4) by $g$ : $\answer {ghx+giy+gjz=gs}$

Add the two new equations from Problems 9 and 10:

$\answer {-jex+ghx-jfy+giy-jgz+gjz=-jr+gs}$

There should be two terms that cancel out to eliminate the $z$ variable.

Multiply Equation (5) by $-j$ : $\answer {-jkx-jly-jmz=-jt}$

Multiply Equation (4) by $m$ : $\answer {mhx+miy+mjz=ms}$

Add the two new equations from Problems 12 and 13:

$\answer {-jkx+mhx-jly+miy-jmz+mjz=-jt+ms}$

There should be two terms that cancel out to eliminate the $z$ variable.

Your equations in Problems 11 and 14 should be linear equations in the variables $x$ and $y$ . These form a $2\times 2$ linear system of equations that you can solve with the equations you came up with in Problems 4 and 8.

Your equations for $x$ and $y$ in Problems 4 and 8 were in terms of the variables $a,b,c,d,p,$ and $q$ . In order to apply these formulas the the $2 \times 2$ linear system formed by the equations in Problems 11 and 14, you need to figure out what $a,b,c,d,p,$ and $q$ are equal to in terms of the variables $e,f,g,h,i,j,k,l,m,r,s,$ and $t$ .

Match your equation from Problem 11 with Equation (1), and match your equation from Problem 14 with Equation (2).

For example, take $a=(gh-je)$ .

Your answer will be much neater if you leave your terms in factored form, i.e., do not FOIL them out.

$x=\answer {\frac {(gs-jr)(mi-jl)-(gi-jf)(ms-jt)}{(gh-je)(mi-jl)-(gi-jf)(mh-jk)}}$ $y=\answer {\frac {(ms-jt)(gh-je)-(gs-jr)(mh-jk)}{(gh-je)(mi-jl)-(gi-jf)(mh-jk)}}$