**8, 14****5, 17****2, 20****4, 18****10, 12**

The aim of the question is to find the value of x and y by solving the given **Simultaneous Equations**.

The basic concept behind the article is the **Solution of Simultaneous Equations**.

**Simultaneous Equations** are defined as a system of equations containing two or more **algebraic equations** having the same **variables** which are related to each other through an equal number of equations. These equations are solved simultaneously for each variable; hence they are called **Simultaneous Equations**.

If we want to solve the given set of two **algebraic equations**, we must find an ordered pair of numbers, which when substituted in the given equations, satisfies both **algebraic equations**.

**Simultaneous equations** are generally represented as given below:

\[ax+by = c\]

\[dx+ey = f\]

Where,

$x$ and $y$ are two **variables**.

$a$, $b$, $c$, $d$, $e$ and $f$ are **constant factors**.

## Expert Answer

Given that:

Let the **first variable** is represented by $x$ and the **second variable** is represented by $y$. The two s**imultaneous equations** based on the relations in the given article will be:

**The First expression of the Simultaneous Equation is:**

The **Second variable** is $2$ more than $3$ times the **First variable**.

\[y\ =\ 2+3x \]

**The Second expression of the Simultaneous Equation is:**

The **sum** of both variables is $22$

\[x+y\ =\ 22 \]

By substituting the value of $y\ =\ 2+3x$ from **First expression** into **Second expression**, we get

\[x+(2+3x)\ =\ 22 \]

\[4x+2\ =\ 22 \]

\[4x\ =\ 22-2 \]

\[4x\ =\ 20 \]

Solving for $x$:

\[x\ =\ \frac{20}{4}\ =\ 5 \]

Hence, the value of **variable** $x$ is $5$.

Now, we will substitute the value of $x=5$ into the **First expression** to calculate the value of **variable **$y$

\[y\ =\ 2+3x \]

\[y\ =\ 2+3(5)\ =\ 2+15 \]

\[y\ =\ 17 \]

Hence, the value of **variable** $y$ is $17$.

## Numerical Result

The numbers corresponding to **variables** $x$ and $y$ for the given set of **simultaneous equations** are

\[x\ =\ 5\ and\ y\ =\ 17 \]

## Example

Find the value of **variables** $x$ and $y$ for the following set of **Simultaneous Equations.**

\[2x+3y\ =\ 8 \]

\[3x+2y\ =\ 7 \]

**Solution**

Given that:

**The First expression of Simultaneous Equations is:**

\[2x+3y\ =\ 8 \]

Solving for $x$

\[2x\ =\ 8-3y \]

\[x\ =\ \frac{8-3y}{2} \]

**The Second expression of Simultaneous Equations is:**

\[3x+2y\ =\ 7 \]

Substituting the value of **variable** $x$ in **second expression**:

\[3\left(\frac{8-3y}{2}\right)+2y\ =\ 7 \]

\[\left(\frac{24-9y}{2}\right)+2y\ =\ 7 \]

\[\frac{24-9y+4y}{2}\ =\ 7 \]

\[\frac{24-9y+4y}{2}\ =\ 7 \]

\[24-9y+4y\ =\ 14 \]

\[9y-4y\ =\ 24-14 \]

\[5y\ =\ 10 \]

\[y\ =\ 2 \]

Now, substituting the value of **variable **$y$ in the expressions for $x$, we get:

\[x\ =\ \frac{8-3y}{2} \]

\[x\ =\ \frac{8-3(2)}{2} \]

\[x\ =\ \frac{2}{2} \]

\[x\ =\ 1 \]

The numbers corresponding to **variables** $x$ and $y$ for the given set of **Simultaneous Equations** are:

\[x\ =\ 1\ and\ y\ =\ 2 \]