To solve a system of equations by addition or subtraction (or elimination), you must eliminate one of the variables so that you could solve for one of the variables. First, in this equation, you must look for a way to eliminate a variable (line the equations up vertically and look to see if there are any numbers that are equal to each other). If there is lets say a –2y on the top equation and a –2y on the bottom equation you could subtract them and they would eliminate themselves by equaling zero. However, this equation does not have any equal terms. So instead we will multiply one or both equations by a number so that they will equal each other resulting in elimination. In this equation we will want to manipulate both equations so that the y’s will both equal –6 (I chose –6 because it is a common term among –2 and –3). Multiply the WHOLE top equation by 3 to equal –6y (you have to multiply 7, 2, and 4 by 3. The outcome of the other two numbers will not matter to the overall equation.) Then multiply the bottom by 2 to equal –6 as well. Again, you have to multiply 2 to the WHOLE equation. Once you finish manipulating the equations you can now eliminate the y variable and only solve for x. If you subtract the 6y’s you must subtract the other numbers from each other as well. After you solve for x, plug it in to any one equation and then solve for y.
...Solvingsystems of linear equations
7.1 Introduction
Let a system of linear equations of the following form:
a11 x1
a21 x1
a12 x2
a22 x2
ai1x1 ai 2 x2
am1 x1 am2 x2
a1n xn
a2 n x n
ain xn
amn xn
b1
b2
bi
bm
(7.1)
be considered, where x1 , x2 , ... , xn are the unknowns, elements aik (i = 1, 2, ..., m;
k = 1, 2, ..., n) are the coefficients, bi (i = 1, 2, ..., m) are the free terms of the system. In
matrix notation, this system has the form:
Ax b ,
(7.2)
where A is the matrix of coefficients of the system (the main matrix), A = [aik]mn, b is the
column vector of the free terms, bT [b1 , b2 , ... , bm ] , x is the column vector of the
unknowns, xT [ x1 , x2 , ... , xn ] ; the symbol () T denotes transposition.
It is assumed that aik and bi are known numbers. An ordered set, {x1, x2, ..., xn}, of real
numbers satisfying (7.1) is referred to as the solution of the system, and the individual
numbers, x1, x2, ..., xn, are roots of the system.
A system of linear equations is:
consistent  if it has at least one solution. At the same time it can be

determined  if it has exactly one, unique solution,
undetermined  if it has infinitely many solutions;...
...Hall
Differential Equations
March 2013
Differential Equations in Mechanical Engineering
Often times college students question the courses they are required to take and the relevance they have to their intended career. As engineers and scientists we are taught, and even “wired” in a way, to question things throughout our lives. We question the way things work, such as the way the shocks in our car work to give us a smooth ride back and forth to school, or what really happens to an object as it falls through the air, even how that people can predict an approximate future population. These questions, and many more, can be answered and explained through different variations of differential equations. By explaining and answering even just one of these questions through different differential equations I will also be answering two other important questions. Why is differential equations required for many students and how does it apply in the career of a mechanical engineer?
First some background. What is a differential equation?
A differential equation is a mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. They are used whenever a rate of change is known but the process
giving rise to it is not. The solution of a differential equation...
...Solving Exponential and Logarithmic Equations
Exponential Equations (variable in exponent position)
1. Isolate the exponential portion ( base exp onent ): Move all nonexponential factors or terms to the other side of the equation. 2. Take ln or log of each side of the equation. • Make sure to use ln if the base is “e”. Then remember that ln e = 1 . • Make sure to use log if the base is 10. • If the base is neither “e” nor “10”, use either ln or log, your choice.. 3. Bring the power (exponent) down into coefficient position. 4. Use various algebra techniques to solve for the variable. 5. Check your answer by evaluating the original equation with your calculator.
Example: 4e 2 x −3 = 40
Answer is on next page
Logarithmic Equations
1. 2. 3. 4. 5. Move all log terms to one side of the equation, all nonlog terms to the other side. Combine log terms into a single log term using the laws of logarithms. Write the log equation in its exponential form. (remember: 2 3 = 8 ↔ log 2 8 = 3 ) Use various algebra techniques to solve for the variable. Check your answer using your calculator. Remember that domain problems occur in log functions. • If the base of the log is “10” or “e”, you can use the appropriate calculator keys. • If the log is not “10” or “e”, you may need to use the change of base formula before using your calculator.
Example:...
...
Equations and ProblemSolving
* An airplane accelerates down a runway at 3.20 m/s2 for 32.8 s until is finally lifts off the ground. Determine the distance travelled before takeoff.

Solutions
Given: a = +3.2 m/s2  t = 32.8 s  vi = 0 m/s 
 Find:d = ?? 
d = VI*t + 0.5*a*t2
d = (0 m/s)*(32.8 s) + 0.5*(3.20 m/s2)*(32.8 s)2
d = 1720 m

Equations and ProblemSolving
* A car starts from rest and accelerates uniformly over a time of 5.21 seconds for a distance of 110 m. Determine the acceleration of the car.

Solutions
Given: d = 110 m  t = 5.21 s  vi = 0 m/s 
 Find:a =?? 
d = VI*t + 0.5*a*t2
110 m = (0 m/s)*(5.21 s) + 0.5*(a)*(5.21 s)2
110 m = (13.57 s2)*a
a = (110 m)/ (13.57 s2)
a = 8.10 m/ s2

Equations and ProblemSolving
* Rocketpowered sleds are used to test the human response to acceleration. If a rocketpowered sled is accelerated to a speed of 444 m/s in 1.8 seconds, then what is the acceleration and what is the distance that the sled travels?

Solutions
Given: vi = 0 m/s  vf = 44 m/s  t = 1.80 s 
 Find:a =??d =?? 
a = (Delta v)/t
a = (444 m/s...
...SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES Solve the following systems: 1.
x y 8 x y 2
by graphing
by substitution
by elimination
by Cramer’s rule
2.
2 x 5 y 9 0 x 3y 1 0
by graphing
by substitution
by elimination
by Cramer’s rule
3.
4 x 5 y 7 0 2 x 3 y 11 0
by graphing
by substitution
by elimination
by Cramer’s rule
CASE 1: intersecting lines independent & consistent m1m2
CASE 2: parallel lines inconsistent m1 = m2 ; b1 b2
CASE 3: coinciding lines consistent & dependent m1 = m2 ; b1 = b2
Classify the following system, whether (a) intersecting, (b) parallel, or (c) coinciding lines 1.
3 x 4 y 1 0 3 x 4 y 2 0 3 x 4 y 1 0 6 x 8 y 2 0
Solve the following systems in three variables: 1. 3 x 4 y z 1 2. x y 2
x 4 y 3z 3 3 x 2 y 2 z 0
________
3 y z 1 x 2 z 7
2.
________
3.
2 x 5 y 1 0 5 x 2 y 2 0
2 x y 1 4 x 2 y 3 x 2 y 1 0 2 x y 1
________
4.
________
5.
________
1 x Solve 1 x
2 3 y 3 2 y
Problem solving Form a system of equations from the problems given below. A) (MIXTURE PROBLEM 1) How many pounds of a 35% salt solution and a 14%...
...1. Which equation below represents the quadratic formula?
*a. b±b24ac2a = x
b. a2+b2=c2
c. fx=a0+n=1∞ancosnπxL+bnsinnπxL
2. Which of the following represents a set of parallel lines?
a. Option one
b. Option two
*c. Option three
3. What is the definition of an obtuse angle?
*a. an angle greater than 90°
b. an angle equal to 90°
c. an angle less than 90°
4. Which formula below represents the area of a circle?
a. A=2πr
*b. A=πr2
c. A=π2r
d. A= √π
5.
What geometric term is represented by the image below?
a. a corner
*b. a crosssection
c. the circumference
d. the perimeter
11. Using the data in the table below, calculate the mean, or average, number of points scored by Player B.
 Game 1  Game 2  Game 3  Game 4  Game 5 
Player A  13  12  9  11  13 
Player B  12  11  15  20  12 
*a. 14
b. 11.5
c. 13
d. 13.67
6. This instrument is commonly used by surveyors. It measures horizontal and vertical angles to determine the location of a point from other known points at either end of a fixed baseline, rather than measuring distances to the point directly. What is it called?
a. triangulator
b. binocular
c. tripod
*d. theodolite
7. What is the name of the missing shape in the flowchart below?
a. Acute
b. Obtuse
*c. Isosceles
d. Right
8. What category includes all of the items on the list below?
* Square
* Rectangle
*...
...While the ultimate goal is the same, to determine the value(s) that hold true for the equation, solving quadratic equations requires much more than simply isolating the variable, as is required in solving linear equations. This piece will outline the different types of quadratic equations, strategies for solving each type, as well as other methods of solutions such as Completing the Square and using the Quadratic Formula. Knowledge of factoring perfect square trinomials and simplifying radical expression are needed for this piece. Let’s take a look!
Standard Form of a Quadratic Equation
ax2+ bx+c=0
Where a, b, and c are integers
and a≥1
I. To solve an equation in the form ax2+c=k, for some value k. This is the simplest quadratic equation to solve, because the middle term is missing.
Strategy: To isolate the square term and then take the square root of both sides.
Ex. 1) Isolate the square term, divide both sides by 2
Take the square root of both sides
2x2=40
2x22= 40 2
x2 =20
Remember there are two possible solutions
x2= 20
Simplify radical; Solutions
x= ± 20
x=± 25
(Please refer to previous instructional materials Simplifying Radical Expressions )
II. To solve a quadratic equation arranged in the form ax2+...
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