SOLVING LINEAR EQUATIONS. If we have a linear equation we can “manipulate” it to get it in this form. We just need to make sure that whatever we do preserves.

Slides:

Advertisements

Similar presentations

Equations in Quadratic Form

Advertisements

< < < > > >         . There are two kinds of notation for graphs of inequalities: open circle or filled in circle notation and interval notation.

Operations on Functions

Solving Quadratic Equations.

Parallel and Perpendicular Lines. Gradient-Intercept Form Useful for graphing since m is the gradient and b is the y- intercept Point-Gradient Form Use.

LINES. gradient The gradient or gradient of a line is a number that tells us how “steep” the line is and which direction it goes. If you move along the.

If a > 0 the parabola opens up and the larger the a value the “narrower” the graph and the smaller the a value the “wider” the graph. If a < 0 the parabola.

PAR TIAL FRAC TION + DECOMPOSITION. Let’s add the two fractions below. We need a common denominator: In this section we are going to learn how to take.

Let's find the distance between two points. So the distance from (-6,4) to (1,4) is 7. If the.

REAL NUMBERS. {1, 2, 3, 4,... } If you were asked to count, the numbers you’d say are called counting numbers. These numbers can be expressed using set.

DOUBLE-ANGLE AND HALF-ANGLE FORMULAS

(r,  ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.

SPECIAL USING TRIANGLES Computing the Values of Trig Functions of Acute Angles.

TRIGONOMETRIC IDENTITIES

You walk directly east from your house one block. How far from your house are you? 1 block You walk directly west from your house one block. How far from.

Logarithmic Functions. y = log a x if and only if x = a y The logarithmic function to the base a, where a > 0 and a  1 is defined: exponential form logarithmic.

INVERSE FUNCTIONS.

The definition of the product of two vectors is: 1 This is called the dot product. Notice the answer is just a number NOT a vector.

Dividing Polynomials.

exponential functions

GEOMETRIC SEQUENCES These are sequences where the ratio of successive terms of a sequence is always the same number. This number is called the common ratio.

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

The standard form of the equation of a circle with its center at the origin is Notice that both the x and y terms are squared. Linear equations don’t.

A binomial is a polynomial with two terms such as x + a. Often we need to raise a binomial to a power. In this section we'll explore a way to do just.

ARITHMETIC SEQUENCES These are sequences where the difference between successive terms of a sequence is always the same number. This number is called the.

LINEAR Linear programming techniques are used to solve a wide variety of problems, such as optimising airline scheduling and establishing telephone lines.

Properties of Logarithms

Logarithmic and Exponential Equations. Steps for Solving a Logarithmic Equation If the log is in more than one term, use log properties to condense Re-write.

A polynomial function is a function of the form: All of these coefficients are real numbers n must be a positive integer Remember integers are … –2, -1,

The Complex Plane; DeMoivre's Theorem. Real Axis Imaginary Axis Remember a complex number has a real part and an imaginary part. These are used to plot.

Library of Functions You should be familiar with the shapes of these basic functions. We'll learn them in this section.

SEQUENCES A sequence is a function whose domain in the set of positive integers. So if I gave you a function but limited the domain to the set of positive.

11.3 Powers of Complex Numbers, DeMoivre's Theorem Objective To use De Moivre’s theorem to find powers of complex numbers.

COMPLEX Z R O S. Complex zeros or roots of a polynomial could result from one of two types of factors: Type 1 Type 2 Notice that with either type, the.

Sum and Difference Formulas. Often you will have the cosine of the sum or difference of two angles. We are going to use formulas for this to express in.

Solving Quadratics and Exact Values. Solving Quadratic Equations by Factoring Let's solve the equation First you need to get it in what we call "quadratic.

Surd or Radical Equations. To solve an equation with a surd First isolate the surd This means to get any terms not under the square root on the other.

COMPOSITION OF FUNCTIONS “SUBSTITUTING ONE FUNCTION INTO ANOTHER”

VECTORS. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude.

Separable Differential Equations. A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Multiply.

Warm Up! Complete the square Quadratic Functions and Models.

Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c). Let’s look.

Dividing Polynomials Using Synthetic Division. List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put.

INTRODUCING PROBABILITY. This is denoted with an S and is a set whose elements are all the possibilities that can occur A probability model has two components:

The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms &

Let's just run through the basics. x axis y axis origin Quadrant I where both x and y are positive Quadrant II where x is negative and y is positive Quadrant.

Solving Trigonometric Equations Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 x y π π 6 -7 π 6 π 6.

We’ve already graphed equations. We can graph functions in the same way. The thing to remember is that on the graph the f(x) or function value is the.

The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms &

(r,  ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.

TRIGONOMETRIC IDENTITIES

Systems of Inequalities.

RATIONAL FUNCTIONS II GRAPHING RATIONAL FUNCTIONS.

THE DOT PRODUCT.

Matrix Algebra.

Absolute Value.

Graphing Techniques: Transformations Transformations Transformations

Operations on Functions

SIMPLE AND COMPOUND INTEREST

Solving Quadratic Equations.

Graphing Techniques: Transformations Transformations: Review

Symmetric about the y axis

exponential functions

Operations on Functions

Symmetric about the y axis

Graphing Techniques: Transformations Transformations: Review

Rana karan dev sing.

Presentation transcript:

SOLVING LINEAR EQUATIONS

If we have a linear equation we can “manipulate” it to get it in this form. We just need to make sure that whatever we do preserves the equality (keeps both sides =) We can add or subtract the same thing from both sides of the equation Notice this is the equation above where a = 3 and b = -3. While this is in the general form for a linear equation, we often want to find all values of x so that the equation is true. You could probably do this one in your head and see that when x = 1 we’d have a true statement 0 = 0 A linear equation in one variable is equivalent to an equation of the form:

We will want to find those values of x that make the equation true by isolating the x (this means get the x all by itself on one side of the equal sign) Since the x is in more than one place and inside of parenthesis the first thing we’ll do is get rid of parenthesis by distributing. Now let’s get all constants (terms without x’s) on the right side. We’ll do this by adding 8 to both sides We are ready to get all x terms on the left side by adding 10x to both sides. + 10x Now get the x by itself by getting rid of the x means 12 times x so we get rid of it by dividing both sides by

Let’s check this answer by substituting it into the original equation to see if we get a true statement. Distribute and multiply Distribute Get a common denominator It checks! ? ? ? ?

Solve the following equation: When we see an equation with fractions, it is generally easiest to find the common denominator and multiply all terms by this common denominator. This will give you an equivalent “fraction free” equation to solve. The common denominator is 12 so we’ll multiply each term by 12. (12) Cancel all denominators Here is our “fraction free” equation to solve Distribute Subtract 4y from both sides and combine like terms Divide both sides by 2

Solve the following equation: Let's get a "fraction free" equation by multiplying by the common denominator. Factor any denominators that will factor so you can determine the lowest common denominator. The common denominator is (x+1)(x-1) so multiply all terms by this Cancel all denominators Here is our “fraction free” equation to solve Distribute Combine like terms subtract 5 from both sides (x+1)(x-1) divide both sides by -1 Make sure the answer would not cause you to divide by 0 in the original equation. (Only 1 and -1 would cause this so we are okay).

Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar