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Fractions in Expressions and 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
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- 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
- Integration of Logarithmic Functions
- 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
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
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- 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
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- 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
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- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
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- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
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- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
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- Washer Method
- Decision Maths
- Geometry
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- Area of Plane Figures
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- Area of Rhombus
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- Composition
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- Convexity in Polygons
- Coordinate Systems
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- Equation of Circles
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- Figures
- Fundamentals of Geometry
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- Glide Reflections
- HL ASA and AAS
- Identity Map
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- Isometry
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- Law of Cosines
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- Linear Measure and Precision
- Median
- Parallel Lines Theorem
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- Plane Geometry
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- Rectangle
- Reflection in Geometry
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- Segment Length
- Similarity
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- Special quadrilaterals
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- Using Similar Polygons
- Vector Addition
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- Volume of Cone
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- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
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- Constant Acceleration
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- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
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- Projectiles
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- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- 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
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- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
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- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
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- Complex Numbers
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- Composition of Functions
- Compound Interest
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- Construction and Loci
- Converting Metrics
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- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
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- Data transformations
- 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
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- 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 Frequency
- Data Analysis
- Data Interpretation
- 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 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
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
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- Scatter Graphs
- Single Variable Data
- Skewness
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

**Algebraic expressions** in mathematics consist of **constants** and **variables** related by **algebraic operations** (additions, subtractions, multiplications, and divisions). A **fraction** is the division of two expressions: for example are fractions of expressions containing constants and / or variables.

An **equation** is a mathematical statement that consists of an **equal symbol between two algebraic expressions**. For example, is an equation.

**Solving an equation **means finding the value of the variable that makes the expressions on the left-hand side and the right-hand side **equal**; this value is called the **solution** of the equation. For example, the solution of the equation is : if you plug in this value on both sides of the equation you get .

This article deals with the presence of **fractions in expressions and equations**; such equations are called (unsurprisingly) **fractional equations.**

The primary step in dealing with fractional equations is to **eliminate the fractions from them**. Let's dive in and see how to achieve this!

**Terms**are the building blocks of algebraic expressions: in an expression, each term is separated by a plus (+) sign or a minus (-) sign. In the equation , 3x and 4 are the terms of the expression on the left-hand side; and 10 is the term of the expression on the right-hand side.**Coefficients**are the values that multiply the variables in either an expression or an equation. Given the expression , the coefficient is that number that multiplies the x, which is.**Variables**are the letters in expressions and equations that are used to represent unknown quantities. When you have an expression given as , x and y both identify as variables.**Constants**are the numbers in expressions and equations that do not change. In the equation for example, 4 stands to be the constant.

If you are dealing with fractions in expressions, it is easiest to add and subtract them when there are **common denominators**. This means that we will find the equivalent fraction of the fractions involved by finding the **Lowest Common Divisor (LCD)** for the denominators of the terms.

Simplify

**Solution:**

What we will do here is to find a common denominator for the two terms so they can be added. Firstly, we will have to find the LCD for the denominators of the two fractions, 5 and 4. The lowest common divisor for both numbers will be 20. Now we will find the equivalent fraction for both.

If the denominator for the first fraction should now be 20, that means we probably multiplied the initial denominator, 5 by 4. This would mean we will have to multiply the numerator also by 4 to have an equivalent fraction.

We will do the same for the second fraction. 20 as a denominator also means that we must have multiplied 4 (as the denominator) by 5 to have 20 as the new denominator of the equivalent fraction. This means we will have to multiply the numerator also by 5.

We now have our new expression:

This becomes a lot easier to solve since all we have to do is add the numerators and maintain the denominators.

Since this cannot be simplified any further, we will leave it at that.

There could be more complicated problems where we may have to use a couple of techniques like **factorizing** and **grouping**. In these situations, one needs to be very careful about what particularly a term is, and when its components are being divided. Let us look at the example below:

Simplify

**Solution:**

Since we cannot cancel anything in this current expression, we may want to factorize to see what we can make of the situation. We will first group like terms in the numerator by rearranging so that terms containing *x* will be close together and terms containing *b* will also be close.

We will now factorize. The common factor in the first two terms in the numerator is x. That can be factored out. The common factor in the last two terms of the numerator is *-b*, and that can also be factored out.

The whole idea of factorizing here is building a common bracket so one can be taken out. Here we have and . Considering the commutative property of addition, both brackets are the same.

This will leave us with:

Now, we will factorize the denominator immediately. As *ax* appears common in both terms, that is what will be factored out.

We are now left with a situation where we can freely cancel out. in the denominator will cancel out in the numerator. This is true if is different from .

This is the simplest form we can get from this expression.

As mentioned earlier, the thing one should have their eyes on when dealing with equations involving fractions is to try to **eliminate the fraction first**. You should multiply all of the terms on both sides of the equation by the fraction's denominator.

If we were given the equation , we would first multiply the equation (which is technically also each term of the equation) by 2.

**Solution:**

After multiplying by 2, the fraction will cancel out.

We will now rearrange the equation to put like terms on different sides of the equation.

Divide both sides by 10

To check that this is indeed the solution of the equation you need to substitute the value of x back into the original equation:

Solve

**Solution:**

An equation with two fractions with the same denominator will have its terms being multiplied with the denominator, as mentioned earlier.

Like terms will now be grouped from this point.

Divide both sides by 4

To evaluate this, you would need to substitute the value of x back into the original equation.

Solve

**Solution:**

Our example is quite different from the usual here. Since we have two fractions with different denominators, we will find the LCM for both and multiply that with the equation. The LCM is 4, so

We will now expand what is in the parenthesis:

Group like terms:

Divide both sides by 13:

To evaluate this, you would need to substitute the value of x back into the original equation.

- The left-hand side and right-hand side of an equation must remain equal when operating on them.
- The primary step in dealing with fractional equations is to eliminate the fractions from them.
- When you have an equation with two fractions and different denominators, find the LCM for both numbers.

1/x +3 is an example of fraction in expressions.

2/3 + 2x = 4 is an example for fractions in equations.

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