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:
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.
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.
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 below is an example of a parabola, which is a U-shaped curve. 
An 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:
A 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).
Graph 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:
Graphical 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).
Graph 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.
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.
, 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.
where a, b and c are all 
