Question

The sum of 2 numbers is 15. The difference between five times the first number and three times the second number is 19. Find the two numbers.​​

Answer 1

The numbers are 8 and 7.

Step by step explanation:

`\textbf{2 x 2 system of equations : }`

➭Approach

Let x and y be the numbers.

`\bold{The \: sum \: of \: 2 \: n \acute{u}mers \: \: is \: 15 \: : \: } \: \bold{x + y = 15}`

• 
  `\bold{x+y=15}` Equation 1

• 
  `\bold{5x-3y=19}` Equation 2

Solution by substitution method

`\begin{gathered}\begin{gathered}\begin{gathered}\begin{cases}{ \bold{x + y= 15 \qquad \cdots(1)}}\\{ \bold{5x - 3y = 19 \qquad \cdots( 2)}} \end{cases}\end{gathered} \end{gathered} \end{gathered}`

In essence, this method of solving consists of clearing an unknown in one of the equations and then substituting the found value in the other equation.

Because it is easier, I will start solving for x in equation 1.

`\bold{x = 15 - y}`

Substitute
`\bold{x=15-y}` into equation 2.

`\bold{5 \cdot15 - y- 3y = 19}`

We are left with an equation with only one unknown, we solve to find y.

`\bold{75 - 5y- 3y = 19}`

`\bold{ - 5y - 3y = 19 - 75}`

`\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{-8y= -56}`

`\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{y= ( - 56)/( - 8)}`

`\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{ \implies{y = 7}} \: \: \: \: \: \: \:`

We have found the value of y, to find the value of x we substitute the value of y into either of the two equations.

`\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{x + y = 15} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{x + 7 = 15 }\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{x = 15 - 7}`

`\bold{\implies \: x = 8}`

『 The attached graph is just to check our results 』

`\therefore` The numbers are 8 and 7.