Solving Equations

数学词汇中英文对照（初中部分）方程与代数数学词汇中英文对照（初中部分）二、方程与代数代数（学）：algebra字母表示数：Use letters to indicate numbers代数式：algebraic expression单项式：monomial系数：coefficient次数：degree多项式：polynomial二项式：binomial三项式：trinomial二次三项式：second degree trinomial项：term常数项：constant term整式：integral expression升幂：in ascending order of the power降幂：in descending order of the power同类项：like terms合并：combine等式：equality, equation等号：sign of equality二次方：(x)squared三次方：(x)cubedn次方：(x)to the power of n/to the n-th power乘法公式：multiplication formula平方差：difference of squares平方差公式：formula for the difference of squares 完全平方：perfect square完全平方公式：formula for the perfect square分解因式：factorizing公因式：common factor提公因式法：method of extracting common factors 十字相乘法：method of cross multiplication分组分解法：method of regrouping长除法：long division分离系数法：method of detached coefficients分式：algebraic fraction无意义：illegal有意义：legal有理式：rational expression约分：reduction of a fraction最简分式：simplest fraction通分：turn fractions to a common denominator最简公分母：simplest common denominator根式：radical根指数：radical exponent被开方数：radicand二次根式：quadratic surd最简二次根式：simplest quadratic surd同类二次根式：similar quadratic surds分母有理化：rationalize a denominator有理化因式：rationalizing factor根：root增根：extraneous root已知数：given number未知数：unknown number方程：equation列方程：form an equation等量关系：equality检验：check根：root解方程：solving equation解法、解：solution一元一次方程：linear equation in one variable方程的解：solution of equation移项：transposition of terms去括号：remove brackes去分母：remove denominator化简：simplify不成立：false不等式：inequality一元一次不等式：linear inequality in one unknown一元一次不等式组：system of linear inequalities in one variable不等号：non-equal sign含绝对值的不等式：inequality with absolute value大于：greater than小于：less than大于等于：greater than or equal to小于等于：less than or equal to不等式性质：property of inequality解集：solution set解不等式：solve inequality公共部分：common part无解：no solution二元一次方程：linear equation in two unknowns二元一次方程组：system of linear equations in two unknowns 代入（消元）法：elimination by substitution加减（消元）法：elimination by addition and subtraction三元一次方程：linear equation in three unknowns三元一次方程组：system of linear equations in three unknowns一元二次方程：quadratic equation in one unknown一般式：general form二次项：quadratic term一次项：linear term常数项：constant term开平方法：radication因式分解法：factorization配方法：complete a perfect squae求根公式法： formula method一元二次方程根的判别式：discriminant of quadratic equation in one variable整式方程：integral equation一元整式方程：linear integral eqution一元高次方程：linear high-order equation二项方程：binomial equation双二次方程：biquadratic equation分式方程：fractional equation无理方程：irrational equation二元二次方程：quadratic equation in two variables一元二次不等式：quadratic inequality in one variable。

SAT 数学知识点一Number and Operations Review 一、Properties of integers知道下列说法表示的内容：1. Integers consist of the whole numbers and their negatives (including zero).2. Integers extend infinitely in both negative and positive directions.3. Integers do not include fractions or decimals.4. Negative integers5. Positive integers6. The integer zero is neither positive nor negative.7. odd numbers（奇数）and even numbers（偶数）8. Consecutive integers9. Addition of integers(奇数偶数的加法规则)10. Multiplication of integers（奇数偶数的乘法规则）二、Arithmetic word problems（算术题）三、Number lines（数轴）四、Square and square roots（平方和平方根）五、Fractions and rational numbers（分数与有理数）六、Elementary number theory☆Factors, multiples, and remainders☆Prime numbers七、Ratios, proportions, and percents八、Sequences九、Sets(union, intersection, elements)十、Counting problems Counting problems involve figuring out how many ways you can select or arrange members of groups, such as letters of the alphabet, numbers or menu selections.☆Fundamental counting problems分步完成事件和分类完成事件发生的可能性☆Permutations and combinations （排列组合）基本排列组合理论十一、Logical reasoningThe SAT doesn’t include1.Tedious or long computations2.Matrix operations1.2.3.4.5. 6.7.8.9.10.11.12.13. 14.SAT数学知识点二Algebra and Functions Review Many math questions require knowledge of algebra. This chapter gives you some further practice. You have to manipulate and solve a simple equation for an unknown, simplify and evaluate algebraic expressions, and use algebraic expressions, and use algebraic concepts in problem-solving situations.For the math questions covering algebra and functions content, you should be familiar with all of the following basic skills and topics:一、Operations on algebraic expressions二、Factoring三、Exponents四、Evaluating expressions with exponents and roots五、Solving equations☆Working with “unsolvable” equations☆Solving for one variable in terms of another☆Solving equations involving radical expressions六、Absolute value 七、Direct translation into mathematical expressions八、Inequalities九、Systems of linear equations and inequalities十、Solving quadratic equations by factoring 十一、Rational equations and inequalities 十二、Direct and inverse variation十三、Word problems十四、Functions☆Function notation and evaluation☆Domain and range☆Using new definitions☆Functions as models☆Linear functions: their equations and graphs☆Quadratic functions: their equations and graphs☆Qualitative behavior of graphs and functions☆Translations and their effects on graphsand functionsThe SAT doesn’t include:一、Solving quadratic equations thatrequire the use of the quadraticformula二、Complex numbers三、Logarithms1.2.3.4. 5.6.7.8.9.10.SAT 数学知识点三Geometry and Measurement Review Concept you should to knowFor the mathematics questions covering geometry and measurement concepts, you should be familiar with all of the following basic skills, topics, and formulas:一、Geometric notation二、Points and lines三、Angles in the plane四、Triangles(including special triangles)☆Equilateral triangles☆Isosceles triangles☆Right triangles and the Pythagorean theorem ☆30º-60º-90ºtriangles☆45º-45º-90ºtriangles☆3-4-5 triangles☆Congruent triangles☆Similar triangles☆The triangle inequality五、Quadrilaterals☆Parallelograms☆Rectangles☆Squares六、Areas and Perimeters☆Areas of squares and rectangles☆Perimeters of squares and rectangles☆Area of triangles☆Area of Parallelograms七、Other polygons☆Angles in a polygon☆Perimeter☆Area八、Circles☆Diameter☆Radius☆Arc☆Tangent to a circle☆Circumference☆Area九、Solid geometry☆Solid figures and volumes☆Surface area十、Geometric perception十一、Coordinate geometry☆Slopes, parallel lines, and perpendicular lines☆The midpoint formula☆The distance formula十二、TransformationsThe SAT doesn’t include:一、Formal geometric proofs二、Trigonometry三、Radian measure1.2.3.4.5.6. 7.8.9.SAT 数学知识点四Data Analysis, Statistics andProbability ReviewFor the math questions covering data analysis, statistics and probability concepts, you should be familiar with all of the following basic skills and topics:一、Data interpretation二、Statistics☆Arithmetic mean☆Median☆Mode☆Weighted average☆Average of algebraic expression☆Using average to find missing numbers三、Elementary probability四、Geometric probabilityThe SAT doesn’t include:四、Computation of standard deviation 1.2.3.4.5. 6.7.8.Word Problems1.2.3.4. 5-75.6.7.1112。

solve - Solve EquationsCalling Sequence:solve(eqn, var)solve(eqns, vars)Parameters:eqn - an equation, inequality or procedureeqns - a set of equations or inequalitiesvar - (optional) a name (unknown to solve for)vars - (optional) a set of names (unknowns to solve for)Description:•The solution to a single equation eqn solved for a single unknown var is returned as an expression. To solve a system of equations for some unknowns the system is specified as a set of equations eqns and a set of unknowns vars. The solutions for vars are returned as a set of equations.•The output from solve in general is a sequence of solutions. In the case of multiple solutions, you should put the solutions in a list or set before manipulating them further. When solve is unable to find anysolutions, the empty sequence NULL is returned. This means that either there are no solutions, or simply that solve was unable to find the solutions. If solve was unable to find some solutions, it will set the global variable _SolutionsMayBeLost to true.•Two shortcuts for the input of equations and unknowns are provided. Wherever an equation is expected, if an expression expr is specified then the equation expr = 0 is understood. If vars is not specified, Maple will solve for all the variables that appear in eqns, i.e. indets(eqns,name) is used in place of vars.•In general, explicit solutions in terms of radicals for polynomial equations of degree greater than 4 do not exist. In these cases, implicit solutions are given in terms of RootOfs. Note that explicit solutions for the general quartic polynomial equation are not by default given because they are too complicated and messy to be of use. By setting the global variable _EnvExplicit to true then solve will return explicit solutions for quartic polynomials in all cases. By setting _EnvExplicit to false, all solutions which are not rational will be reported as RootOfs.•The number of solutions found can be controlled by changing the value of the global variable _MaxSols.If an integer is assigned to _MaxSols, only those many solutions will be computed and returned. Thevariable _EnvAllSolutions, if set to true, will force all inverse transcendental functions to return the entire set of solutions. This usually requires additional, system created, variables, which take integer values.Normally such variables are named with prefix _Z for integer values, _NN for non-negative integervalues and _B for binary values (0 and 1).•A single quadratic equation with constant coefficients in one variable is solved directly by substitution into the quadratic formula. However, if _EnvTryHard is set to true, Maple will try to express solutions in the common radical basis. This may lead to a nicer answer, but can take a long time. •When abs is used in the context of solve, solve assumes that the argument to abs is real-valued.•For solving differential equations, use dsolve . For purely floating-point solutions use fsolve . Use isolve to solve for integer solutions, msolve to solve modulo a prime; rsolve for recurrences, and linalg[linsolve] to solve matrix equations. •Further information is available for the subtopics solve[subtopic] where subtopic is one offloat function identity ineq linear radical scalar series system•For systems of polynomial equations, the function Groebner[gsolve] which uses a Grobner-basis approach may be useful.Examples:> solve( f=m*a, a );f m> solve( {f=m*a}, {a} );{}=a f m > f := proc(x) x-cos(x) end proc:solve( f(x),x);()RootOf − _Z ()cos _Z > eq := x^4-5*x^2+6*x=2;:= eq = − + x 45x 26x 2> solve(eq,x);,,,111−1> sols := [solve(eq,x)];:= sols [,,,111−1> sols[1];1> evalf(sols);[],,,1.1..732050808-2.732050808> sols := {solve(eq,x)};:= sols {},1−11> sols[1];1> subs( x=sols[1], eq );= 22> solve(x^4+x+1,x);()RootOf , + + _Z 4_Z 1 = index 1()RootOf , + + _Z 4_Z 1 = index 2,,()RootOf , + + _Z 4_Z 1 = index 3()RootOf , + + _Z 4_Z 1 = index 4,> evalf({%});− .7271360845.9340992895I − -.7271360845.4300142883I ,,{ + -.7271360845.4300142883I + .7271360845.9340992895I ,}Manipulating Solutions: solve for u,v,w> eqns := {u+v+w=1, 3*u+v=3, u-2*v-w=0};:= eqns {},, = + 3u v 3 = − − u 2v w 0 = + + u v w 1> sols := solve( eqns );:= sols {},, = v 35 = w -25 = u 45check solutions> subs( sols, eqns );{},, = 33 = 11 = 00pick off one solution> subs( sols, u );45assign all solutions> assign( sols );u;45Other Examples> solve( cos(x)+y = 9, x );− π()arccos − y 9> solve( x^2+x>5, x );, RealRange ,−∞ Open − − 12RealRange , Open − + 12∞> solve( x^6-2*x^2+2*x, x );0()RootOf , − + _Z 52_Z 2 = index 1()RootOf , − + _Z 52_Z 2 = index 2,,,()RootOf , − + _Z 52_Z 2 = index 3()RootOf , − + _Z 52_Z 2 = index 4,,()RootOf , − + _Z 52_Z 2 = index 5> solve( {x^2*y^2=0, x-y=1} );,,,{}, = y 0 = x 1{}, = y 0 = x 1{}, = x 0 = y -1{}, = x 0 = y -1> solve( {x^2*y^2=0, x-y=1, x<>0} );,{}, = y 0 = x 1{}, = y 0 = x 1> solve( {x^2*y^2-b, x^2-y^2-a}, {x,y} );{}, = y ()RootOf + − _Z 4_Z 2a b = x ()RootOf − − _Z 2()RootOf + − _Z 4_Z 2a b 2a> _EnvAllSolutions := true:solve( sin(x)=cos(x)-1, x );,−+ 12π2π_Z12π_Z2> r := solve(cos(x),x);:= r + 12ππ_Z3~> subs(op(indets(r))=3, r); # substitute 3 for _Z3~ in r72πSee Also:RootOf , allvalues , dsolve , fsolve , isolve , msolve , rsolve , assign , invfunc , isolate , match , linalg[linsolve], simplex , Groebner。

「高中数学全面知识点梳理与重要公式汇总」Title: Comprehensive Review and Important Formula Compilation of High School MathematicsAbstract:In this article, we will provide a comprehensive review of the key knowledge points and important formulas in high school mathematics. The aim is to help students consolidate their understanding and enhance their problem-solving skills in this subject. The article will cover various topics including algebra, geometry, trigonometry, calculus, and statistics. Through a combination of English and Chinese explanations, we hope to create a useful resource for students to revise and prepare for exams.Introduction:High school mathematics is a fundamental subject that lays the groundwork for further studies in science, technology, engineering, and mathematics (STEM) fields. It is crucial for students to have a solid understanding of the key knowledgepoints and be familiar with the important formulas. This article aims to serve as a comprehensive guide, providing a thorough review of these concepts.I. Algebra:1. Basic operations: Addition, subtraction, multiplication, and division of real numbers.2. Equations and inequalities: Solving linear equations and inequalities.3. Functions: Understanding the concept of a function, graphing linear and quadratic functions.4. Exponents and logarithms: Laws of exponents, properties of logarithms.5. Polynomials: Factoring, long division, synthetic division.6. Rational expressions: Simplifying, multiplying, dividing rational expressions.7. Systems of equations: Solving systems of linear equations using substitution, elimination, and matrices.二、几何:1. 基本概念: 点、线、面、角度、直线、平行线、垂直线、多2. 三角形: 三角形的分类、三角形的性质、勾股定理、正弦定理、余弦定理等。

方程的英语知识点总结Key Concepts of Equations:1. Definition of an Equation: An equation is a mathematical statement that asserts the equality of two expressions, typically denoted as LHS = RHS, where LHS (left-hand side) and RHS (right-hand side) are mathematical expressions containing variables and constants.2. Variables and Constants: In an equation, variables are symbols that represent unknown quantities, while constants are fixed numerical values. Equations allow us to solve for the value of the variable by manipulating the given information and applying various mathematical operations.3. Solutions of an Equation: The solution of an equation is the value or set of values for the variable that make the equation true. A solution to an equation satisfies the equality relationship between the LHS and RHS.4. Solving Equations: The process of finding the solutions to an equation involves using algebraic techniques to manipulate the given expressions and isolate the variable. Common methods for solving equations include combining like terms, applying inverse operations, and factoring.5. Equivalent Equations: Two equations are said to be equivalent if they have the same solution set. Algebraic manipulations such as adding or subtracting the same quantity from both sides, multiplying or dividing both sides by the same non-zero number, and applying the properties of exponents can be used to derive equivalent equations.6. Applications of Equations: Equations are used to model various real-world scenarios, such as calculating the trajectory of a projectile, determining the growth of populations, analyzing the behavior of electrical circuits, and predicting the spread of infectious diseases. Types of Equations:1. Linear Equations: A linear equation is an equation of the form ax + b = c, where x is the variable, a and b are constants, and c is a constant term. The graph of a linear equation is a straight line, and the solutions to a linear equation form a single point, a line, or no points (in the case of parallel lines).2. Quadratic Equations: A quadratic equation is an equation of the form ax^2 + bx + c = 0, where x is the variable, and a, b, and c are constants with a ≠ 0. Quadratic equations have solutions that can be found using the quadratic formula, factoring, or completing the square. The graph of a quadratic equation is a parabola.3. Exponential Equations: An exponential equation is an equation in which the unknown variable appears as an exponent. Exponential equations arise in situations involving exponential growth or decay, such as population growth, radioactive decay, and compound interest problems.4. Trigonometric Equations: Trigonometric equations involve trigonometric functions such as sine, cosine, and tangent. These equations often arise in problems related to periodic phenomena, wave functions, and harmonic motion.Properties of Equations:1. Reflexive Property: For any real number a, a = a.2. Symmetric Property: If a = b, then b = a.3. Transitive Property: If a = b and b = c, then a = c.4. Addition Property of Equality: If a = b, then a + c = b + c.5. Subtraction Property of Equality: If a = b, then a - c = b - c.6. Multiplication Property of Equality: If a = b, then ac = bc.7. Division Property of Equality: If a = b and c ≠ 0, then a/c = b/c.8. Multiplicative Property of Zero: For any real number a, a × 0 = 0.9. Multiplicative Property of One: For any real number a, a × 1 = a.10. Distributive Property: For any real numbers a, b, and c, a(b + c) = ab + ac.In conclusion, equations are a vital aspect of mathematics and are used to express and solve a wide range of problems in various fields. Understanding the key concepts, types, and properties of equations is essential for mastering algebra and applying mathematical principles to real-world situations. By studying equations and their properties, one can develop problem-solving skills and analytical thinking, which are invaluable in academic, professional, and everyday life.。

我想要数学的作文英语Mathematics is like a puzzle that never ends. It challenges our minds and pushes us to think in new ways. Whether it's solving equations, calculating probabilities, or exploring geometric shapes, math is always full of surprises.Numbers are the building blocks of math. They can be added, subtracted, multiplied, and divided to create endless possibilities. From counting apples to measuring distances, numbers help us make sense of the world around us.Geometry is all about shapes and space. It's about understanding the relationships between lines, angles, and surfaces. Whether it's finding the area of a circle or constructing a triangle, geometry helps us see the beautyin the world through a mathematical lens.Algebra is like a secret code waiting to be cracked. Itinvolves using symbols and variables to represent unknown quantities. By solving equations and inequalities, we can unlock the mysteries hidden within the numbers and find solutions to complex problems.Statistics is the art of making sense of data. It involves collecting, analyzing, and interpreting information to make informed decisions. From calculating averages to predicting trends, statistics helps us understand the world around us in a quantitative way.Mathematics is not just about numbers and formulas.It's about creativity, problem-solving, and critical thinking. It's about seeing patterns where others see chaos and finding beauty in the complexity of the world. So let's embrace the challenge of math and explore the endless possibilities it has to offer.。

方程英语知识点总结1. What is an equation?An equation is a mathematical statement that shows the equality of two expressions. It typically contains one or more variables and is used to express a relationship between these variables. For example, the equation 2x + 3 = 7 shows that the expression 2x + 3 is equal to 7 when x has a value of 2.2. Types of equationsThere are several types of equations, each with its own unique characteristics and methods for solving. Some of the most common types of equations include:- Linear equations: These are equations of the form ax + b = c, where a, b, and c are constants, and x is the variable. Linear equations have a constant slope and form a straight line when graphed on a coordinate plane.- Quadratic equations: These are equations of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. Quadratic equations have a squared term and form a parabola when graphed.- Polynomial equations: These are equations that contain terms with multiple powers of the variable, such as x^3 + 2x^2 - 5x + 3 = 0.- Exponential equations: These are equations that involve exponential functions, such as2^x = 8.- Logarithmic equations: These are equations that involve logarithmic functions, such as log(x) = 2.- Trigonometric equations: These are equations that involve trigonometric functions, such as sin(x) = 0.5.- Rational equations: These are equations that contain rational expressions, such as 1/(x-2) + 1/(x+3) = 1.Each type of equation has its own specific methods for solving, and they can be used to model various real-world phenomena and solve practical problems.3. Solving equationsSolving an equation involves finding the values of the variable(s) that satisfy the given equation. There are several methods for solving different types of equations, including:- Algebraic methods: These include techniques such as factoring, completing the square, or using the quadratic formula for solving quadratic equations.- Graphical methods: These involve graphing the equation on a coordinate plane and determining the points of intersection with the x-axis to find the solutions.- Numerical methods: These involve using iterative algorithms or numerical techniques to approximate the solutions of equations that cannot be solved algebraically.- Trigonometric identities: These are used to simplify and solve trigonometric equations by using the properties of trigonometric functions.- Logarithmic and exponential properties: These are used to simplify and solve equations involving logarithmic and exponential functions by applying the properties of these functions.4. Applications of equationsEquations have numerous applications in various fields, and they are used to model and solve real-world problems in science, engineering, economics, and many other disciplines. Some common applications of equations include:- Physics: Equations such as Newton's second law of motion (F = ma) or the laws of thermodynamics are used to describe and predict the behavior of physical systems.- Engineering: Equations are used to design and analyze structures, machines, and electrical circuits, as well as to solve problems related to fluid mechanics, heat transfer, and materials science.- Economics: Equations are used to model economic relationships and solve optimization problems in areas such as supply and demand, production functions, and cost analysis.- Chemistry: Equations such as the ideal gas law (PV = nRT) or the rate laws for chemical reactions are used to describe the behavior of chemical substances and reactions.- Biology: Equations are used to model population growth, genetics, and ecological systems, as well as to analyze biological data and make predictions about biological processes.These are just a few examples of how equations are used in various fields, and they demonstrate the importance of equations in understanding and solving real-world problems.In conclusion, equations are a fundamental concept in mathematics, and they are used to represent relationships between variables and solve a wide range of problems in different fields. By understanding the different types of equations, methods for solving them, and applications in real-world problems, students can develop a strong foundation in mathematics and gain valuable skills for their future careers.。

学习数学的英语作文English:Studying mathematics is not merely about memorizing formulas and solving equations; it is a journey of exploration and discovery that opens doors to critical thinking and problem-solving skills. Mathematics provides a universal language that transcends cultural barriers, enabling communication and collaboration on a global scale. Through the study of mathematics, one develops analytical reasoning and logical deduction abilities, which are essential in various fields such as science, engineering, finance, and technology. Moreover, mathematics cultivates perseverance and resilience as students encounter challenges and setbacks, encouraging them to persist and find innovative solutions. Beyond its practical applications, mathematics stimulates creativity and imagination, fostering a sense of wonder and awe for the beauty and elegance inherent in its patterns and structures. Embracing the study of mathematics not only enriches one's academic and professional pursuits but also nurtures a lifelong appreciation for the power and elegance of abstract reasoning and problem-solving.Translated content:学习数学不仅仅是记忆公式和解方程，它是一次探索和发现之旅，打开了批判性思维和解决问题技能的大门。