If you are studying algebra, you need to be rather good at solving equations. The ability to solve equations is crucial for you to understand this subject. While some equations are straight forward like 2x=4, other equations like 3x+1= 4x-5 can be more complicated to solve. A solution (or solution set) is the way in which you can solve any algebraic equations. What is a solution set and how is it used in equations? Let us find out.
What Is a Solution Set?
A solution set for any equation is any number (or set of numbers) that can be simply plugged into the equation variable to solve the problem.
As an example, consider the following simple equation
2x -4=8
Here is how you would solve this equation:
2x -4+4=8+4
2x=12
x=122=6
Or the value of “x” is 6. Substituting the value of x as “6” would solve the original equation. In other words, the solution set for the equation 2x -4=8 is denoted as {6}.
Other equations can also have more than one solution variable as the answer. In which case, the solution set can have more values, represented as {x1, x2, x3, …}. For example, for the following equation
a= a2 the variable “a” can have values 0 or 1 (as the square of only these two numbers is equal to the number itself). In this case, the solution set is represented as {0,1}.
In other cases, solution sets can also have an infinite number of values or it can also be empty. Here are the solution sets for various types of equations:
| Equation Example | Solution Set |
| 2x -4=8 | {6} |
| a= a2 | {0,1} |
| a+1=1+a | {R} or the infinite set of all real numbers |
| x2= -25 | ᶲ also known as the empty set (as the square of no number can be a negative value) |
Solution sets can also include negative numbers. For example, in the equation a= a2, the solution set can also include -1.
Solution Set for Inequalities
An inequality is any statement that denotes the relation between 2 or more expressions using operators including < (less than), ≤ (less than or equal to), > (more than), ≥ (more than or equal to), or ≠ (not equal to).
Just like equations, you can define an inequality using a variable and then plug in a value to the variable to determine when the expression is true. For example, consider the following expression
5y+4<10
Let us simplify it first as follows:
5y+4-4<10-4
5y<6
would be true when the value of “y” is either 0 or 1.
Solution sets can also work for inequalities and can include a single value or many values depending on the complexity of the inequality. In the above simple inequality example, the solution set would be {0,1}. Similarly, the solution set for an inequality statement like 0x4 would be {0,1,2,3,4}.
Let us now consider a more complex inequality statement as shown below:
y+4> -6
Solving this equation, we would get:
y+4-4> -6-4
y> -10
Here the value of the variable, “y” can be any number greater than -10 or an infinite number of values. Hence, the solution set is denoted as {y|y> -10} where the pipe (or | sign) means “such that”. So, you would read this solution set as “y such that y is greater than negative ten”.
Interval Notation
Solution sets for inequalities can also be represented using a mathematical concept called interval notation. How does it work? Imagine a real number line with the plotted numbers ranging from -5 to 5. An interval notation can be used to write subsets of the overall number line.
Primarily, you can use the following types of interval notations:
- Closed interval that includes both its endpoints. Example, {y|-4 ≤ y ≤ 1}. For this interval, we need to use closed brackets (in this case, [-4,1]).
- Open interval that does not include its endpoints. Example, {y|-4 < y < 1}. For this interval, we need to use parentheses (in this case, (-4,1)).
What Are Replacement Sets?
A replacement set is a set of multiple values that can be used to find a solution to an equation. For example, determine the solution set for the following equation:
20-5x= -5
using the replacement set {0, 5, 10}.
Try to replace each of the replacement set values as “x” in the above equation, as follows:
20-5 x 0= -5
20 -5 (hence “0” is not a valid value)
20-5 x 5= -5
20-25= -5
-5= -5 (hence “5” is a valid value)
20-5 x 10= -5
20-50= -5
-30≠ -5 (hence “10” is not a valid value)
The only value that fits the solution would be 5, hence the solution set is represented as {5}.
On the other hand, consider this equation x2= x with the replacement set {0,1,2,3}. Only the values 0 and 1 would solve this equation, so the solution set would be {0,1}.
Replacement Sets and Inequalities
Replacement sets also work for inequalities by plugging in values that can make the inequality statement true.
For example, consider a simple inequality statement:
a-5>10
with the replacement set {5, 10, 15, 20, 25, 30}.
As before, plug in each of the replacement values into the inequality statement as follows:
5-5>10 (false)
10-5>10 (false)
15-5>10 (false)
20-5>10 (true)
25-5>10 (true)
30-5>10 (true)
The solution set for this statement would be {20,25,30} as only these values for “a” would satisfy the above statement.
On a similar note, replacement sets can also include negative numbers. For example, apply this replacement set {-5,-4,-3,-2,-1,0,1} to the statement -3x≥9 to get the solution set as {-5,-4,-3}.
Solution Sets and Linear Equations
Linear equations are any equation that uses multiple variables (example, x, y, z) or a constant number. Here are some examples of linear equations:
x+2y=50
2x+3y-5z=10
Linear equations can also include a set of multiple linear equations (or a system of equations) such as the following:
{a+2b+3c=14 2a-3b+2c=2 3a+b+c=8
For such a system of equations, a solution set would comprise values for the variables (in this case, a, b, c) that would satisfy each of the individual equations. For the above, the solution set would be {1,2,3} as the values 1 (for “a”), 2 (for “b”), and 3 (for “c”) would make all the three equations true.
In other words, for a system of equations with “n” number of variables, a solution set would always contain a list of “n” numbers.
Just like other equations, a system of equations need not always have a solution set or in other words, an empty solution set. A system of equations that do not have a solution are referred to as inconsistent. Here is an example:
{x+y=5 x+y= -5
Every system of equations that does have a solution is referred to as consistent.
Solution Sets and Matrix Equations
A system of linear equations can also be represented in a concise manner using a matrix equation, as follows:
Ax=b
- Where A is a matrix (represented as m x n) with “m” rows and “n” columns.
- x and b are vectors with different sizes.
To simplify this, consider A as a matrix with v1, v2, v3,..vn columns.
| | | v1 v2 vn | | |
The value of A with vector “x” (with values x1, x2, …xn) would be calculated as:
Ax= x1v1+ x2v2+…+ xnvn
To make sense of Ax, “x” should have the same number of entries (or “n”) as the number of columns in matrix A. Here we are using the “x” entries as coefficients for the columns of A in the linear equation.
Consider a vector equation with the vectors v1, v2, v3,..vn and b as shown below:
x1v1+ x2v2+…+ xnvn=b
This is same as writing Ax=b. On the other hand, if A is an m x n matrix, then Ax=b would be equal to the above vector equation. In other words, the vector equation x1v1+ x2v2+…+ xnvn=b and the matrix equation Ax=b will have the same solution set.
The matrix equation Ax=b will have a solution if the following conditions are satisfied:
- x1, x2, x3, …xn exist such that A x1x2 x3 xn =b
- x1, x2,…xn exists such that x1v1+ x2v2+…+ xnvn=b
- b is a linear vector of v1, v2, v3, …vn.
- b is in the span of “n” or the number of columns in A.
As discussed in this article, solution sets can be used in a wide variety of algebra formula and examples including equations, inequality statements, linear and matrix equations and much more. A solution set can consist of a single or multiple value, infinite number of values, or even null values.
We have also learned the mathematical symbols that are used to denote a solution set. Can you try solving some algebraic equations and arriving at the correct solution set? All the best in tackling your mathematical problems.
Article links:
https://www.varsitytutors.com/hotmath/hotmath_help/topics/solution-sets
https://www.sparknotes.com/math/algebra1/expressionsandequations/section3
https://tutorial.math.lamar.edu/Classes/Alg/SolutionSets.aspx
