Equations

Jul 09, 2014 240 likes | 617 Views Solving Systems of Equations. Vocabulary. System of Equations: a set of equations with the same unknowns Consistent : a system of equations with at least one ordered pair that satisfies both equations. Download Presentation Solving Systems of Equations An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher. Presentation Transcript Solving Systems of EquationsVocabulary • System of Equations: a set of equations with the same unknowns • Consistent: a system of equations with at least one ordered pair that satisfies both equations. • Dependent: a system of equations that has an infinite number of solutions; (lines are the same) • Inconsistent: a system of equations with no solution • Independent: a system of equations with exactly one solution • Solution to a system of equations: a point that makes both equations trueDifferent Methods for Solving Systems • Elimination Method: adding or subtracting the equations in the systems together so that one of the variables is eliminated; multiplication might be necessary first • Substitution Method: Solving one of a pair of equations for one of the variables and substituting that into the other equation • Graphing Method: Solving a system by graphing equations on the same coordinate plane and finding the point of intersection.Substitution Method STEPS: • Solve one of the equations for one of the variables • Substitute, or replace, the resulting expression into the other equation for the solved variable • Solve the equation for the second variable • Substitute the found variable into either

83x + 2PurposeEstablishes a relationship between two expressions or values, showing they are equivalent.Represents a value or combination of values that can be evaluated or simplified.SolvingCan be solved by finding the values of variables that make both sides equal.Can be simplified by performing arithmetic or algebraic operations.UseUsed in mathematical problems, modeling relationships, and real-world applications to find unknown values.Often used to create equations or to simplify and evaluate values.Important Notes on Equations in Math:Equations are a fundamental part of mathematics and have numerous applications across various fields. Here are some key points to consider:Defining Equality: Equations assert that two expressions are equal, creating a balanced relationship between variables, constants, and terms.Types: Equations come in various types, including linear, quadratic, cubic, polynomial, exponential, logarithmic, rational, trigonometric, and differential equations. Each type has unique properties and applications.Graphical Representation: Many equations can be visualized graphically, such as linear equations forming straight lines, quadratic equations creating parabolas, and cubic equations producing curves with multiple turning points.Solving Methods: Different equations require different solving techniques, such as algebraic manipulation, factoring, completing the square, the quadratic formula, Cardano’s method, or logarithmic transformation.Applications: Equations are used in various fields, including physics, engineering, economics, and computer science. They model relationships, such as distance-time-speed in physics, profit functions in economics, and algorithms in computer science.Systems of Equations: Multiple equations can form systems, which can be solved simultaneously to find common solutions. This is particularly useful in modeling complex relationships between variables.Real and Complex Solutions: Depending on the type of equation,

Arithmetics? Review the basics of arithmetics calculations and learn about representation of integers and real numbers using our interactive solvers. Graphing equations Graphing equations with our interactive graphers is easy! Not only do they let you graph equations, but with our tutorials you also learn about characteristic shapes and points of of common equations. Exponential function Is exponential function exponentially harder to learn than other functions? Not so with our interactive solvers, graphers and tutorials! Graphing inequalities If you already know how to graph equations, inequalities are easy-peasy. Lots of shading involved! Use our interactive graphers and tutorials to learn this important topic Linear equations in two variables This set of tutorials will explain how to solve and graph a system of two linear equations in a variety of ways Radical equations There is nothing very radical about radical equations! That is, if you know what a radical is. If not, go through our Roots and Radicals tutorial first. Then come back and use our interactive radical equations solvers and tutorials to learn how to solve this type of equations. Simplification of expressions This set of tutorials explains how to simplify a variety of expressions, such as fractions and radicals. Linear inequalities Good news: Solving linear inequalities inequalities is very similar to solving linear equations. Bad news: there is this pesky little inequality sign () that can change directions based on what you do with your inequality. Read our tutorials and use the absolute value solver to learn more.

Ensure they satisfy the original equation.Types of EquationsEquations can be classified into various types based on their structure, degree, and the nature of their variables. Here are some key types of equations:Linear Equations: These are equations of the form ax + b = 0 where a and b are constants, and x is a variable. They graph as straight lines and have a degree of one.Quadratic Equations: These equations are of the form ax² + bx + c = 0 where a, b, and c are constants, and x is a variable. They graph as parabolas and have a degree of two.Polynomial Equations: These have the general form aⁿxⁿ + aⁿ⁻¹xⁿ⁻¹ + ... + a¹x + a⁰ = 0 where n is the degree of the polynomial, and each term includes a constant coefficient a and a power of x.Exponential Equations: In these equations, the variable appears as an exponent, such as in aˣ bˣ = c. Solving these often involves logarithmic functions.Logarithmic Equations: These involve logarithmic functions, such as log_b(x) = y, and can be rewritten into exponential form for easier manipulation.Rational Equations: These include rational expressions, such as fractions with polynomials in the numerator and denominator. An example is (x+1)/(x-1) = 3.Trigonometric Equations: These involve trigonometric functions like sine, cosine, or tangent, such as sin(x) = 0.5. Solutions often require knowledge of inverse trigonometric functions or identities.Differential Equations: These involve derivatives, such as dy/dx = 3x. They describe the relationship between a function and its rate of change,

Unduly influenced by any overrepresented subpopulations. This is a much larger population than the samples used to develop other popular BMR equations using height, weight, and age, such as the Harris-Benedict equation (239 subjects), the revised Harris-Benedict equation (337 subjects), and the Mifflin-St Jeor equations (498 subjects).Since the Oxford/Henry equations were developed, a meta study found that the Oxford/Henry equations had the best combination of low error (small average deviations between measured and predicted BMRs) and low bias (not systematically over- or under-estimating BMR in particularly large or particularly small people) across both sexes. Similarly, another huge study with nearly 17,000 subjects found that the Oxford/Henry equations were among the best-performing equations for people in all BMI categories. Finally, a 2022 meta-analysis found that the FAO/WHO/UNU equations performed best in people with overweight and obesity, but that review included relatively few studies that used the Oxford/Henry equations. However, BMR estimates provided by the FAO/WHO/UNU equations and the Oxford/Henry equations tend to converge at higher body weights and BMIs (in other words, if the FAO/WHO/UNU equations perform well in people with obesity, the Oxford/Henry equations do too).Overall, in most populations, the Oxford/Henry equations are the best BMR equations based on height, weight, age, and sex.Cunningham, 1991Much like the Oxford/Henry equations, the 1991 version of the Cunningham equation is the result of synthesizing data from multiple other studies. Cunningham had first developed an equation for estimating BMR from fat-free mass in 1980. In the intervening decade, more research groups investigated the relationship between fat-free mass (FFM) and BMR, allowing Cunningham to systematically analyze the results from a total population of 1482 subjects. The studies included both males and females, with a pretty even mix of lean and obese subjects.Subsequent research has supported the validity of the 1991 Cunningham equations. For instance, a decade after Cunningham’s study, Wang and colleagues analyzed the FFM/BMR relationship in the published research (which included another seven studies that came out after Cunningham’s equation was published). It was a somewhat less rigorous analysis – I don’t believe they applied weightings based on the number of subjects in each study – but it found that the “average” equation to predict BMR from FFM was BMR = 21.5 * FFM + 407, which is practically indistinguishable from Cunningham’s equation.Furthermore, they modeled the theoretical relationship between FFM and BMR that was revealed from animal research spanning a range of 7 orders