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One number is 2 more than 3 times another. Their sum is 22. Find the numbers.

  • 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 simultaneous 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:

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

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