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Equations

Equations

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

Algebraic and numerical equations

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:

What are polynomial 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

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

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

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.

What are linear equations?

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.

Equations Equations Linear Equations Graph StudySmarter An example of a linear equation graph, Nicole Moyo - StudySmarter Originals
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.

The graph of a quadratic equation is called a parabola. The equation of a parabola is generally written as . The graph below is an example of a parabola, which is a U-shaped curve.

Equations Graph showing a parabola Study SmarterAn example of a parabola - StudySmarter Originals

What are simultaneous equations?

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

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:

Equations Linear Simultaneous Equation Graph StudySmarterA graphical representation of a linear simultaneous equation - StudySmarter Originals

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).

Equations Graph Solution of Linear Simultaneous Equation Study SmarterGraph solution of linear simultaneous equation, - StudySmarter Originals

Quadratic simultaneous equations

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:

Equations Quadratic Simultaneous Equation Example StudySmarterGraphical interpretation of quadratic simultaneous equation -StudySmarter

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).

Equations Graph Solution of Quadratic Simultaneous Equation Study SmarterGraph solution of quadratic simultaneous equation -StudySmarter Originals

Equations - Key takeaways

  • 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.

Frequently Asked Questions about Equations

An equation is a mathematical statement with an equal sign that equates two algebraic or numerical expressions. 

The difference is that algebraic expressions have variables, whilst numerical expressions only have numbers.

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

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

 A linear equation is a special type of polynomial, where all variables and terms are raised to a power of 1. 


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.



Final Equations Quiz

Question

What is the difference between equations and expressions?  

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Answer

An equation is a mathematical statement that consists of an equal sign whilst an expression is a mathematical phrase with no equal sign. 

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Question

What is the difference between numerical and algebraic expressions? 

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Answer

Algebraic expressions consist of variables whilst numerical expressions only consist of numbers. 

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Question

What conditions differentiate an equation from being a polynomial?

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Answer

The conditions that make an Equation a non-polynomial are: 

  • Fractional Exponents: 
  • Radicals: 
  • Division by a variable: 
  • Negative Exponents: 

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Question

State the first three types of polynomial equations and their conditions. 

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Answer

The first three types of polynomial equations are: 

  • Linear polynomial equations, which are of degree 1. 
  • Quadratic polynomial equations, which are of degree 2. 
  • Cubic polynomial equations, which are of degree 3. 

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Question

What is a linear equation?

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Answer

A linear equation is a special type of polynomial equation where all variables or terms are raised to a power of 1. 

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Question

What is the standard form of a linear equation with two variables? 

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Answer


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Question

How do we rearrange into ?

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Answer

Rearrange  to have the y variable stand on its own, on one of the equation sides. 

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Question

Determine the slope and intercept of the following equation: 

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Answer


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Question

Roughly sketch a linear equation graph. 

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Answer

Linear equation graphs are always straight lines 

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Question

What is a quadratic equation? 

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Answer

A quadratic equation is when the degree of the equation is 2.

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Question

Write out the standard form of a quadratic equation and its important condition. 

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Answer


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Question

What are the roots of a quadratic equation? 

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Answer

The roots of a quadratic equation are its x-intercepts and what solves the equation. 

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Question

How can we solve quadratic equations, i.e. find the roots of the equation? 

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Answer

We can solve quadratic equations in one of the following ways: 

  • Factoring: Determining the factors of a quadratic expression.
  • Completing the square: Rearranging the standard form into a perfect square trinomial.
  • Quadratic Formula: 

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Question

Roughly sketch a quadratic equation graph. 

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Answer

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Question

How are linear simultaneous equations solved? 

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Answer

Linear Simultaneous Equations are solved in one of the following ways: 

  • Elimination 
  • Substitution 
  • Graphically

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Question

How are quadratic simultaneous equations solved? 

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Answer

Quadratic simultaneous equations are solved through the process of elimination by substitution and graphically. 

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