# Solving Systems of Linear Equations

## Ohms Law

An element with resistance R is called a resistor. Under most conditions, resistors can be approximate
a linear model . Ohms law relates the voltage and current of resistor, v =i R. Ohms law can also be
expressed in terms of conductance . The conductance G is the inverse of resistance. The units of

conductance are Siemens (S). Thus G =1/R and i =G v.

Using Ohms law we can obtain two other forms for the power dissipated by a resistor.

## Nodes and Loops

An electric circuit is one or more circuit elements connected together. Junctions where two or more circuit
elements are connected together are called nodes. A loop is a closed path that does not include any node
more than once.

The circuit of Figure 2 has three nodes: a, b, and c and three loops: V1 – R1 – R2 – V1, V1 – R1 – R3 –
V1, and R2 – R1 – R2.

## KCL and KVL

Kirchoff’s current law (KCL): The algebraic sum of the currents entering a node in a circuit at any instant
in time is zero.

Kirchoff’s voltage law (KVL): The algebraic sum of the voltages around any loop in a circuit is zero for all
time
.
For the circuit of Figure 3, there are three nodes a, b, and c. Writing the KCL equations for each node we
obtain (assuming currents follow passive convention with exception of current through source V1 and

Writing the KVL equations for each node we obtain ( negative sign indicates voltage drop is from – to +
terminal)

Typically one does not need all possible KCL and KVL equations to solve for the voltages and currents in
a resistive circuit. One needs enough KCL and KVL equations that when combined with Ohms law yields
the same number of equations and unknowns.

## Solving Systems of Linear Equations

A system m linear equations in n unknowns can be written in the matrix form Ax = b, where A is a
m×n matrix of coefficients , x is an n×1 column vector of unknowns, and b is a m×1 column
vector of constants. The matrix A is square when the number of equations and unknowns are equal
(m = n) .

When the A matrix is square and has a inverse, then the system of equations has a unique solution and the
unknowns
can be found from x=A-1b
Using Ohms law, some of the KCL and KVL equations, and v1 =10V for the circuit of Figure 2, we
obtain the following seven equations in seven unknowns

Which we could solve, but substituting the equations from Ohms law into the KVL equations yields four
equations in four unknowns (the currents) which is simpler to solve . The voltages can then be found by
substituting the solutions for the currents back into the Ohm’s law equations.

 Prev Next

Start solving your Algebra Problems in next 5 minutes!

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of April 19th you will pay only \$39.99 instead of our regular price of \$74.99 -- this is \$35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1\$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:

Algebra Helper can solve problems in all the following areas:

• simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
• factoring and expanding expressions
• finding LCM and GCF
• (simplifying, rationalizing complex denominators...)
• solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
• solving a system of two and three linear equations (including Cramer's rule)
• graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
• graphing general functions
• operations with functions (composition, inverse, range, domain...)
• simplifying logarithms
• basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
• arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!