Facebook
Google
GitHub
LinkedinNetwork Analysis Techniques
Simultaneous Equations for Circuit Analysis
25 questions By Tony R. Kuphaldt
-
Question 7 of 25
Suppose you were given the following two equations and asked to find solutions for x and y that will satisfy both at the same time:
y + x = 8 y − x = 3 If we manipulate the second equation so as to solve for y, we will have a definition of y in terms of x that we may use for substitution in the first equation:
y = x + 3 Show the process of substitution into the first equation, and how this leads to a single solution for x. Then, use that value of x to solve for y, resulting in a solution set valid for both equations.
Reveal answerIf y + x = 8 and y = x + 3, then (x + 3) + x = 8. Therefore,
x = 2.5 and y = 5.5 Notes:This question demonstrates one of the (many) practical uses of algebraic substitution: solving simultaneous systems of equations.
-
Question 8 of 25
An interesting and useful property in mathematics is the transitive property:
If a = b and b = c , then a = c Simply stated, two variables must be equal to one another if they are both equal to a common (third) variable. While not particularly profound or breathtaking in scope, this property is nevertheless useful in solving certain mathematical problems.
Suppose you were given the following two equations and asked to find solutions for x and y that will satisfy both at the same time:
y + x = 8 y − x = 3 Manipulate both of these equations to solve for y, and then explain how you could apply the transitive principle to solve for x.
Reveal answerIf 8 − x = y and 3 + x = y, then 8 − x must equal 3 + x:
8 − x = 3 + x Solutions for x and y:
x = 2.5 and y = 5.5 Notes:This method of solving for a two-variable set of simultaneous equations is really nothing more than substitution in disguise. Some students find it easier to grasp than straight substitution, though.
-
Question 9 of 25
Suppose you were given the following two equations and asked to find solutions for x and y that will satisfy both at the same time:
y + x = 8 y − x = 3 Now, you know that we may do anything we want to either equation as long as we do the same thing to both sides (on either side of the “equal” sign). This is the basic rule we follow when manipulating an equation to solve for a particular variable. For example, we may take the equation y + x = 8 and subtract x from both sides to yield an equation expressed in terms of y:
Following the same principle, we may take two equations and combine them either by adding or subtracting both sides. For example, we may take the equation y − x = 3 and add both sides of it to the respective sides of the first equation y + x = 8:
What beneficial result comes of this action? In other words, how can I use this new equation 2y = 11 to solve for values of x and y that satisfy both of the original equations?
Reveal answerWe may use the result (2y = 11) to solve for a value of y, which when substituted into either of the original equations may be used to solve for a value of x to satisfy both equations at the same time.
Notes:While not intuitively obvious to most people, the technique of adding two entire equations to each other for the purpose of eliminating a variable is not only possible to do, but very powerful when looking for solutions to satisfy both original equations. Discuss with your students why it is allowable for us to add y − x to y x and to add 3 to 8. to yield the equation 2y = 11.
