* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download A Review of Linear Eq. in 1 Var.
Two-body Dirac equations wikipedia , lookup
Maxwell's equations wikipedia , lookup
Two-body problem in general relativity wikipedia , lookup
Debye–Hückel equation wikipedia , lookup
Unification (computer science) wikipedia , lookup
BKL singularity wikipedia , lookup
Schrödinger equation wikipedia , lookup
Perturbation theory wikipedia , lookup
Dirac equation wikipedia , lookup
Van der Waals equation wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
Euler equations (fluid dynamics) wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Equations of motion wikipedia , lookup
Calculus of variations wikipedia , lookup
Itô diffusion wikipedia , lookup
Equation of state wikipedia , lookup
Derivation of the Navier–Stokes equations wikipedia , lookup
Differential equation wikipedia , lookup
Schwarzschild geodesics wikipedia , lookup
Note to interview committee: These notes represent a typical 1 hour 10min. lecture on linear equations. It would follow a previous 1 hour 10 min. on the addition and multiplication properties of equality Linear Equations in 1 Variable & The Addition & Multiplication Prop. of Equality Recall: Linear Equations in One Variable are equations that can be simplified to an equation with one term involving a variable raised to the first power which is added to a number and equivalent to a constant. Such an equation can be written as follows: ax + b = c a, b & c are constants a0 x is a variable Our goal is to rewrite a linear equation into an equivalent equation (an equation with the same solution set) in order to derive a solution set (the answer; note that I will simply call it a solution at this point). We will do this by forming the equivalent equation: x=# or #=x x is a variable # is any constant Recall: The Addition Property of Equality says that given an equation, if you add (subtract) the same thing to both sides of the equal sign, then the new equation is equivalent to the original. A = B is the same as A + C = B + C (since C is added to both A & B) We use the addition property to move terms across the equal sign, so that we can get all variable terms on one side and all constant terms on the other side. The addition property works on terms that have been added(subtracted) to(from) other terms. Example: 5x + 3 = 4x 4 Recall: The Mulitiplication Property of Equality says that two equations are equivalent if both sides have been multiplied by a nonzero constant. a = b is equivalent to ac = bc a, b & c are real number c 0 We use the multiplication property to “remove” the numeric coefficient (a number that is multiplied by a variable). It should only be applied only once, after the addition property has been applied. 2 Example: /3 x = 14 Y. Butterworth Linear Equations in 1 Variable 1 Today we will cover solving algebraic equations that will require simplification, at least one application of the addition property as well as the multiplication property. The following will be our goals: 1) Review the process for solving an algebraic equation in one variable 2) Solve problems with ascending degree of difficulty using the step-through process 3) Discuss clearing an equation of fractions/decimals, and solve problems that involve clearing 4) Discuss the 3 types of equations and their solutions Solving Linear Equations in One Variable Step 1: Simplify both sides of the equation a) Use distributive property to simplify complex terms b) Clear the equation of fraction/decimals c) Combine like terms on left and like terms on right Step 2: Step 3: Move all variables to one side of the equation using addition prop. Move all constants to the opposite side of the equation from variables using addition prop. Remove numeric coefficient of variable by multiplying by its reciprocal in order to solve the equation (multiplication prop.) Check your solution Step 4: Step 5: Example: a) Solve each of the following linear equations in one variable. We will check the last one only as a review for checking. 12x 5 = 23 2x Note: At this time I would allow the students time to work the following two examples on their own while walking around the classroom to answer questions. I would allow 3-5 minutes for this process, after which time I would quickly solve the examples as you will see me do. b) 2x + 10 + 14x = 2x 18 c) 5(2x 1) 6x = 4 Now we need to spend a little extra time working on one particular step in the simplification process. Although the steps in simplifying can be done in any order, and some people prefer to clear before distributing, I find that it is better to distribute before clearing for the beginning student. Fewer errors are made in doing the steps in the following manner. Simplifying: Getting Rid of Fractions (Clearing the Equation of Fractions) Step 1: Distribute (although this is not necessary, I think less errors are made this way) Step 2: Find LCD of all fractions (on both left and right) Step 3: Multiply all terms by LCD a) b) c) Multiply symbolically Cancel Multiply out Step 4: Simplify left & right sides of the equation Y. Butterworth Linear Equations in 1 Variable 2 Example: a) Solve by clearing the equation of fractions before using the addition and multiplication properties. 1 /2 x 6/7 = 1 b) 4 c) x /3 (5 w) = - w /2 1 = x/5 + 2 Simplifying: Getting the Decimals Out Step 1: Distribute (although this is not necessary, I think less errors are made this way) Step 2: Examine all of the decimals in the equation, and determine which has the most decimal places. Count those decimal places. Step 3: Multiply every term in the entire equation by a factor of 10 with the same number of zeros as the number counted in step 1. Step 4: Simplify both sides of the equation Example: a) b) Solve by clearing each equation of decimals first. 2.7 x = 5.4 4(y + 2.81) = 2.81 Note: In a problem involving the distributive property, it may be less confusing for you to simplify the distributive property before doing the first step. c) Y. Butterworth 0.60(z 300) + 0.05z = 0.70z 0.41(500) Linear Equations in 1 Variable 3 Finally, we need to discuss equations that have no solution or infinitely many solutions. An equation that has no solution yields an untrue statement. In other words, when you go through the process of solving such an equation you will get a false statement such as 0 = 5. This is a contradiction (an equation with no solution) and its solution set is written as a null set (symbolized with or { }). An equation with infinitely many solutions yields a true statement. In other words, when you go through the process of solving such an equation you will get a true statement such as 5 = 5. This is called an identity (an equation with infinite solutions) and its solution set is all real numbers. I will allow you to write . The following is a little table that shows all the possibilities. Type Conditional Contradiction Identity Example 2x + 5 = 5x 5 2x + 5 = 2(x 5) 2x + 5 = 2(x 5) + 15 Example: a) When Solved Looks Like Solution x = 0 x = 0 or {0} 0 = -5 or { } 0=0 Solve the following equations 2x + 5 = 8x 2(3x + 5) b) 2(x 5) = 8x 2(3x + 5) c) 2x + 5x 10 = 8x 2(3x + 5) Note: Watch the conditional equations that have a solution of zero! If you move the constants before you move the variables, it is easy to think that you have a solution that yields a false statement! If you do the problem by moving the constants first you will get a variable expression that equals a different variable expression. This can be mistaken as a false statement but you can not yet determine this because all the variables are not on one side and the constants on the other! Before you can see the solution, the variable must be on one side and the constants on the other!! For part c), you would have gotten 7x = 6x which does not yield a false statement! Note: I will make copies of my notes to hand out to my students for sections that require a lot of writing, such as word problems, but the notes are always available to students on my web-site. You can find notes similar to these at on my Algebra page under Ch. 2 notes either for Lial or Martin-Gay at http://hhh.gavilan.edu/ybutterworth/algebra/algebraindex.htm. Y. Butterworth Linear Equations in 1 Variable 4