Contents

**Solution|Definition & Meaning**

**Definition**

The term **“solution”** is used in mathematics to refer to a **specific value** or **set of values** that **satisfy** a particular **equation or set of equations**. **Solving** an **equation** of a system of equations **involves** **finding** a **solution** to the equation.

**Overview of Solution**

Figure 1 – Overview of solution

Suppose a **scenario** where we have **two variables** and one of them let’s say **x represents four boxes** and **y represents** an **unknown** number of **boxes,** **both** of the** variables** **sum** up **to give** a total of **9 boxes**. We can write this problem in equation form as:

x + y = 9

As y = 4

4 + y = 9

We want to **find** the **solution** to the equation **by finding** the **value of y** that **satisfies** the **equation**.

Rewriting the equation:

4 + y = 9

y = 9 – 4

y = 5

Putting in the original equation :

4 + (5) = 9

9 = 9

So the **solution** for the above equation that** satisfies** the **equation** is** y = 5**.

This is an **example** of an **analytic solution**, which is a **solution** that is expressed **in terms of mathematical operations** and **functions**. In this case, the **solution** was** found by** performing **algebraic operations** on the **equation** to **isolate** the **variable y**.Â

**Methods for Solution**

There are many **different methods** that can be used to **solve equations** and systems of equations, **depending** on the **type of equation** and the complexity of the problem. Some of the most common methods include **graphing, substitution, elimination, and factoring**.

**Graphing Method**

Figure 2 – Solution of equations through the graphical method

A technique for solving linear equations and systems of linear equations is graphing. In order** to solve** an equation **by graphing** it, one must **first plot** the **equation** **on** a coordinate **plane** and then **search** for the **locations** where the** x or y-axis** **intersects** the equation. The **equation’s solutions** are r**epresented** by **these points**.

**Substitution Method**

Figure 3 – Eliminating and Substituting to get the solution of the equation

A technique for solving linear equations and systems of linear equations is substitution. When **using substitution** to solve an equation, we** first answer** one **equation** for **each variable in terms of the other equation**, and then we **insert this solution** into the **different equation**. This **enables** us **to solve for** the **other variable** while also removing one of the variables.

**Elimination Method**

Systems of linear equations can be solved using the elimination technique. In order **to eliminate** one of the variables from a system of equations, we **add or subtract the equations**, and **then** we **solve** for the **remaining variable**. This approach is **comparable to substitution**, but in some circumstances, it **may be more effective**.

**Factoring Method**

**Polynomial equations** can be **solved** using the **factoring technique**. We **first construct** the** equation** in **standard form**, with the **highest-order term** on the **left** and the **constant term** on the **right**, in order **to factor** the **equation**. The **left side** of the **equation** is then **attempted** to be **factored into** the **sum of the two polynomials**. In the event that we are successful, we can **next resolve** the **equation** by **putting** all of the **factors** at **zero** and **calculating** the** roots**.

**Types of Solution**

There are several types of solutions in mathematics:

**Analytic Solutions**

**Solutions** that can be **described** in terms of **mathematical operations** and functions are referred to as analytical solutions. Typically, **equations** or systems of equations must be **solved** in order **to find** an a**nalytical solution**.

**Numerical Solutions**

These are **solutions** that have been **roughly calculated** using **numerical techniques** like **finite differences** or **Monte Carlo simulations**. When **analytic solutions** are either **not possible** or too **hard to find**, **numerical solutions** are frequently **used**.

**Graphical Solutions**

**Visually portrayed solutions**, such as graphs or diagrams, are referred to as graphic solutions. To **illustrate relationships** between **variables or to see patterns** and trends in data, graphical solutions can be utilized.

**Exact SolutionsÂ **

**Solutions** that are exact, these are solutions that are **exact and precise**. As **opposed** to **numerically** **estimating** the solution, **exact solutions** are often **found by solving equations** or systems of equations.

**Approximate Solutions**

**Solutions** that are **close enough** to be** useful but are not exact** in nature are called **approximate solutions**. When an **exact solution** is **not required** or when it is** too challenging** to **achieve an exact solution**, **approximate solutions** are frequently **utilized**.

**Implicit Solutions**

Implicit solutions are those that are **indicated** **by** the **situation or problem** but are **not stated clearly**. Making assumptions about the issue or **manipulating** the **available information** can **lead to the discovery** of **implicit solutions**.

**Steps to Find a Solution**

To find the solution to an equation, you can follow these steps:

**Identification of the ProblemÂ **

**Identifying the problem** that you seek to solve is the first step toward finding the solution. It is** necessary to read** the **problem statement**, identify the **variables involved**, and** understand** the **context** in which the problem is framed.

**Define GoalÂ **

**Determine your goal** after identifying the problem. In this case, it may be necessary to **determine** a **relationship between variables**, or to solve a **particular unknown** in order to find a value.

**Gathering InformationÂ **

The next step is to** gather** any **information** or **data** that you will need to solve the problem. You may need to **gather data**, **identify equations** and **formulas**, or** make assumptions** about the problem.

**Identification of the EquationÂ **

To determine the solution to an equation, you must **identify** the **equation** that** describes** the **problem**. An **equation** can be **formed** **by** using **formulas** or **relationships** or **manipulating** the problem conditions.

**Solving the Equation**

Now that you’ve found the equation that describes the problem, you can **use** **algebraic** **techniques** to solve it. Methods such as **factoring, completing squares, and quadratic formulas** can be used.

**Verifying Your SolutionÂ **

Once you have come up with a** solution**, you should** verify** that you have come up with a **reasonable** and **accurate solution**. Test your** solution** with **additional data** or **verify** that it **meets** the **problem conditions**.

**Communicating Your ResultsÂ **

Be **concise and clear** with your results, using appropriate language and **notation**.Â

**Solved Examples**

**Example 1**

For example, consider the **equation 2x + 5 = 9. **Find the solution.

### Solution

To solve this equation, you could follow these steps:

**Identification of the problem**: You are trying to find the value of x that makes the equation 2x + 5 = 9 true.**Define Goal**: You are trying to find the value of x.**Gathering information**: You are given the equation 2x + 5 = 9.**Identification of the equation**: This is the equation you are trying to solve.**Solving the equation**: You can use a**lgebraic techniques**to isolate the variable x on one side of the equation. In this case, you can**subtract 5****from both sides**of the equation to get**2x = 4**. Then, you can divide both sides of the equation by 2 to get x = 2.**Verifying your solution:**You can**check your solution**by**substituting x = 2**back into the**original equation**to verify that it is a valid solution.**Communicating your results**: You can communicate your results by stating that the solution to the equation is x = 2.

**Example 2**

Consider **three circles** as **variable y** and **unknown circles** as **variable x** that are **added** to give the **result** of **9**. Find the solution of the equation that satisfy the equation.

**Solution**

The exact solution to the problem is shown below.

Figure 4 – Example of the exact solution

So the exact solution that satisfies the equation is **x = 3**.

*All mathematical drawings and images were created with GeoGebra.*