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# Simultaneous Equations|Definition & Meaning

## Definition

“**Simultaneous**” means to occur at the **same** place or time. Simultaneous equations are defined as a **group** of equations with a common set of **variables**. They are also known as a** system of equations** and are used together to find a solution for** unknown** variables.

## Simultaneous Linear Equations

Simultaneous **linear** equations have all the variables with a **power**/degree of **one**. Consider the** two** simultaneous **equations** given below:

5x – 4y = 21

3x + 5y = 20

Both equations have the **variables x** and **y** common in them.

The solution of the two equations is the **values** of **(x, y)** which** satisfy** both the linear **simultaneous** equations. **Figure 1** shows the graph for the above equations.

In **figure 1**, point **(5,1)** is the **solution** of the above simultaneous equations.

### Point of Intersection

The point of **intersection** is where the two linear** simultaneous** equations or lines **cross** each other. This is the solution that **satisfies** both simultaneous equations.

In **figure 1**, the point **(5,1)** is also known as the **point of intersection**.

A set of **two** simultaneous equations with two variables **x** and **y** have their **point** of intersection in the **x-y plane**.

Similarly,** three** simultaneous equations with three variables will have an **intersecting point** or solution in the** x-y-z space**.

## Simultaneous Equations in Matrix Form

**Simultaneous **equations can also be written in **matrix **form. Consider the **three** simultaneous equations given below

2x – 5y +4z = 10

3x + 7y – z = 16

6x + 3y +8z = 20

**Figure 2** shows these simultaneous equations in **matrix** form.

The matrix equation in **figure 2** can also be written as:

CX = D

Where **C** is the **co-efficient** matrix and **D** is the **variable** matrix. The system is solved for the variables **x**,**y**, and **z**.

Through** matrices**, the simultaneous equations can be** solved** by using the above **equation** as follows:

X = $C^{-1}$.D

Where **$C^{-1}$** is the **inverse** of matrix **C**. It is obtained by using the **formula** given below:

\[ C^{-1} = \frac{adj \ C}{det \ C} \]

Where “**adj C**” is the adjoint of **C** and “**det**” stands for determinant.

## Methods To Solve Simultaneous Equations

The simultaneous equations are solved by finding a **solution** that **satisfies** all the equations.

To solve the equations, the **number** of **variables** in the equations should be equal to the number of simultaneous **equations**.

The following **methods** are used for solving simultaneous equations.

### Elimination Method

In the **elimination** method, a variable from **two** simultaneous equations is **removed** or eliminated to obtain a **one-variable** equation.

This is done by **adding** or **subtracting** the equations with the same coefficients of the variable being eliminated.

The **addition** is performed when the **coefficients** of the variable have **opposite** **signs** and** subtraction** is done when the co-efficient signs are the **same** to eliminate the variable.

**Multiplication** can also be done on an equation to have the same **coefficient** of the eliminating variable.

### Substitution Method

The substitution method is **convenient** when the equations are not **adding** or **subtracting** to **cancel** out a variable.

In this method, **one** of the two simultaneous **equations** is written in such a way with the substitution **variable** on one side and the rest of the **terms** on the other side.

This is then **substituted** into the second **equation** to form a **one-variable** equation. It is solved for the variable’s** value**.

A set of **three** simultaneous **equations** will have the substitution process **twice**.

### Graphing

The two linear simultaneous equations can also be **plotted** on the graph to acquire the **solution**.

Some **values** of **y** are calculated from the equations with known **x** values. The **points** are plotted to get two **lines** for the two **linear** equations. The lines coincide at a point **(x,y)** which is the required **solution**.

## Types of Solutions

Simultaneous equations can have the following **types** of solutions.

### One Solution

Two simultaneous **equations** can have a **single** solution if there is one **point** of **intersection** between the two lines. It means that there is only **one point** that satisfies both equations.

**Figure 3** shows two equations that have one solution.

The solution is **(2,1)**.

### No Solution

Simultaneous equations can also have **no solution** if they are **not crossing** each other at some point. These linear equations are **parallel** to each other.

**Figure 4** shows the plot of two equations with no solution.

### Infinite Solutions

There can be **infinite** solutions to simultaneous equations. This occurs when one equation is multiplied by a **common multiple** to form the other equation.

**Figure 5** shows **infinite** solutions of the following** simultaneous** equations given below:

4x + y = 7

16x + 4y = 28

Notice that the first equation is **multiplied** by **4** to get the second equation. These two **lines** are one over the other on the graph.

## Example Problem of Simultaneous Linear Equations

Consider the **simultaneous** equations:

**8x + 3y = 108** … eq-1

**10x – 15y = -90** … eq-2

Find the solution by using the **elimination method**. Also, show the point of **intersection** on the graph.

### Solution

To **eliminate y**, the first equation is **multiplied** by **5** to have **y** with the same coefficients.

5( 8x + 3y = 108 )

40x + 15y = 540 … eq-3

**Adding** equations 2 and 3 gives:

10x – 15y + 40x + 15y = -90 + 540

50x = 450

With y eliminated, the **value of x** can be found by **dividing** both sides by **50**:

**x = 9**

To find the **value of y**, put the value of **x** in equation 1:

8x + 3y = 108

8(9) + 3y = 108

72 +3y = 108

72 – 72 +3y = 108 – 72

3y = 36

**Dividing** by **3** into both sides gives the value of** y** as:

**y = 12**

So, the** solution** of the simultaneous equations is **(9, 12)**. **Figure 6** shows the plot of simultaneous equations and their intersecting point.

*All the images are created using GeoGebra.*