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Equations

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
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- Arithmetic Series
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- Candidate Test
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- Continuity
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- Cost and Revenue
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- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
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- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
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- Limits at Infinity and Asymptotes
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
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- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
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- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
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- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
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- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Power
- Projectiles
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- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
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- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Spearman's Rank Correlation Coefficient
- Standard Deviation
- Standard Error
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

An equation is a mathematical statement that defines the equality between two expressions. These expressions can either be algebraic or numerical.

An **algebraic expression** consists of variables, constants or coefficients and algebraic operations (addition, subtraction, multiplication and division).

Here are examples of algebraic expressions:

And some examples of algebraic equations:

A **numerical expression** only consists of numbers and algebraic operations.

Here are examples of numerical expressions:

And some examples of numerical equations:

A polynomial equation consists of multiple terms formed from variables, positive integer exponents, coefficients, and only the following algebraic operations: addition, subtraction or multiplication. An equation is not a polynomial when it consists of a radical, negative exponent, divided variable or fractional exponent.

Identifying the difference between polynomial equations depends on the degree of the polynomial. The degree means the value of the largest variable exponent in the equation.

We will focus on the first three types and most common polynomial equations:

Linear polynomial equations are when the degree of the equation equals 1.

For example:

is a linear polynomial equation as the largest variable exponent is 1, i.e.

Quadratic polynomial equations are when the degree of the equation equals 2.

For example:

is a quadratic polynomial equation as the largest variable exponent is 2, i.e.

Cubic polynomial equations are when the degree of the equation equals 3. For example:

is a cubic polynomial equation as the largest variable exponent is 3, i.e.

There is a vast number of polynomial equation names, so other equations with a degree higher than 3 can be referred to as just a polynomial equation with the standard form being:

An example of a polynomial equation with a degree greater than 3 is which has a degree of 7, i.e. . Fun fact: this would be referred to as a **septic polynomial equation**.

Linear equations are a special type of polynomial equation. It describes when all variables or terms are raised to a power of 1.

For example:

The standard form of linear equations changes according to the number of variables included, i.e. if there is one variable there will be one coefficient, if there are two variables, there will be two coefficients, etc. To solve a linear equation, we need to determine the value of the variables.

The standard way to write a linear equation with one variables is .

The standard way to write a linear equation with two variables is , where m is the gradient intercept and c is the y-intercept. For example, whereby and . This form is also referred to as the slope-intercept form.

Another way of writing linear equations with two variables is where a, b and c are all real numbers. An example of this form is:

Let's have a look at how to solve linear equations:

**Step 1:** Simplify both sides of the equation, by multiplying out any parentheses.

**Step 2:** Rearrange (by adding or subtracting) the equation to have all the like terms be on the same sides of the equation.

**Step 3: ** Use multiplication or division to solve the equation by determining the value of the variable.

The graph below is an example of a linear equation graph which is always an infinite straight line.

What are quadratic equations?

Quadratic equations are another special type of polynomial equation and are defined as an equation of second degree.

For example:

The standard form of a Quadratic Equation is where a, b and c are all real numbers and . The coefficient of can never equal 0 as this would transform the standard form into a linear equation.

Quadratic equations consist of roots, which are the x-intercepts of the equation. These solve the equation. They can be calculated in one of the following ways:

Factoring: finding the terms that multiply together to get a mathematical expression.

Completing the Square: rearranging the standard form of a quadratic equation into a perfect square trinomial with a constant, , on the left side.

Quadratic Formula: finding the solution of a quadratic equation, by using its coefficients and constant term. The quadratic formula is:

Let's solve a quadratic equation by using one of the methods above – the quadratic formula:

**Step 1: ** List out the values of a, b and c.

**Step 2:** Substitute these values into the quadratic formula and solve for the roots/solutions.

Simultaneous Equations are a set of two or more algebraic equations that also share two or more unknown variables. We solve these equations together to calculate the values of these unknown variables, which would be the only pair that solve all of the equations at the same time.

The most common simultaneous equations are linear simultaneous equations and quadratic simultaneous equations. Let's have a look at the distinction between these two:

Linear simultaneous equations are when two or more linear polynomial equations need to be calculated simultaneously. We can solve them through one of the following processes: elimination, substitution or graphically.

An example of a linear simultaneous equation is:

A graphical interpretation of a simultaneous equation would indicate how many solutions the equation has. The number of intersections represents the number of solutions. In most cases, linear simultaneous equations only have one solution. See below:

Let's have a look at how to solve a basic simultaneous linear equation through the process of substitution:

**Step 1: ** Label each equation, as you please. In this example, our equations will be labelled as 1 and 2.

**Step2: ** Simplify one of the equations to have one of the variables on its own.

Note that it is still equation 1, just simplified.

**Step 3:** Substitute the value of the separated variable into the other equation, ie: replace y with 2x in equation 2 and then solve the equation.

**Step 4:** Substitute the value of the calculated variable x, into the simplified equation (equation 1), to determine the value of the separated variable y.

The graph below shows how this equation would be sketched, where the one solution would be (2.4).

A quadratic simultaneous equation is when there is at least one quadratic polynomial equation involved in calculating equations simultaneously. We can solve them graphically, and through the process of elimination by substitution.

An example of a quadratic simultaneous equation is:

As previously mentioned, the number of intersections on a simultaneous equation graph tells us how many solutions the equation has. Most quadratic simultaneous equations have two solutions. See below:

Let's have a look at how to solve a basic quadratic simultaneous equation through the process of elimination by substitution:

**Step 1: ** Eliminate the variables by substituting the simpler equation into the other and then solve the equation to find the value of one variable. In our example, the second equation is simpler.

**Step 2:** Find the value of the remaining variables by substituting the calculated variables into the simpler equation (equation 2).

The graph below shows how this example would be sketched, where the solutions are (1,4) and (-5,-2).

The main types of polynomial equations are linear, quadratic and cubic polynomial equations, which depend on the degree of the polynomial.

- Polynomial equations with a degree higher than 4 can be referred to as just polynomial equations.
The most common standard form of a linear equation is .

The standard form of a quadratic equation is .

The most common simultaneous equations are linear simultaneous equations and quadratic simultaneous equations.

Examples of numerical expressions are: 22+3, 10, 30-2.3/2

Examples of algebraic expressions are: 2a-1, 3², w-23

The roots of a quadratic equation can be calculated by:

- Factoring or factorising: Finding the terms that multiply together to get a mathematical expression.
- Completing the square: Rearranging the standard form of a quadratic equation into a perfect square trinomial with a constant, a(x+d)^2+e=0, on the left side.
- Quadratic formula: Finding the solution of a quadratic equation, by using its coefficients. The quadratic formula is: x=-b+-sqrt(b²-4ac)/2a

Balancing equations means making sure that both sides of an equation are always equal. Therefore, what you do on one side you do on the other, i.e. if you add or subtract, multiply or divide a number, term, coefficient, etc on one side then you would do the same on the other side. Not doing the same thing on both sides of the equation, is a common mistake made when solving equations. Let's have a look at an example:

-6x-2=-15x+34

-6x-2+15x=-15x+15x+34

-6x-2+15x=34

-6x+15x-2+2=34+2

-6x+15x=34+2

As seen, our equation is kept balanced by first adding the term on both sides and further adding 2 on both sides. When solving the equation the same concept needs to be followed. Please see below:

-6x+15x=34+2

9x=36

9x/9=36/9

x=4

As seen, our equation is kept balanced and solved by dividing 9 into both sides.

More about Equations

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