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The sum of two numbers is 37. Their difference is 13 fine the smaller number​

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[tex] \large\underline{\mathbb{PROBLEM}:} [/tex]

  • The sum of two numbers is 37. Their difference is 13, find the smaller number.

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[tex] \large\underline{\mathbb{ANSWER}:} [/tex]

[tex] \qquad \large \:\rm{12 \: is \: the \: smaller \: number} [/tex]

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[tex] \large\underline{\mathbb{SOLUTION}:} [/tex]

» Let x and y be the larger and smaller number respectively. Create two equations by the given situations.

  • [tex] \begin{cases} x + y = 37 \\ x - y = 13 \end{cases} \quad \begin{align} \tt{(eq. \: 1)} \\ \tt{(eq. \: 2)} \end{align} [/tex]

» Find x in the first equation the substitute it to the second equation in terms of y.

  • [tex] \begin{cases} x = 37 - y \\ x - y = 13 \end{cases} [/tex]

  • [tex] \begin{cases} x = 37 - y \\ 37 - y - y = 13 \end{cases} [/tex]

  • [tex] \begin{cases} x = 37 - y \\ 37 - 2y = 13 \end{cases} [/tex]

  • [tex] \begin{cases} x = 37 - y \\ 2y = 37 - 13 \end{cases} [/tex]

  • [tex] \begin{cases} x = 37 - y \\ 2y = 24 \end{cases} [/tex]

  • [tex] \begin{cases} x = 37 - y \\ 2y/2 = 24/2 \end{cases} [/tex]

  • [tex] \begin{cases} x = 37 - y \\ y = 12 \end{cases} [/tex]

[tex] \therefore [/tex] The value of y or the smaller number is 12.

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(ノ^_^)ノ

The smaller number is 12.

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Let [tex]m[/tex] and [tex]n[/tex] be the two numbers, to have the system of equations:

[tex]\begin{cases}m+n=37 \\ m-n=13 \end{cases}[/tex]

Add both equations

[tex](m+n)+(m-n)=37+13[/tex]

[tex]2m=50[/tex]

[tex]\displaystyle \therefore m=\frac{50}{2}=\boxed{25}[/tex]

Thus, one of the numbers is 25. Now, substitute [tex]m=25[/tex] to the first equation and solve for the other number [tex]n.[/tex]

[tex]m+n=37[/tex]

[tex]25+n=37[/tex]

[tex]\therefore n=37-25=\boxed{12}[/tex]

We are asked to find the smaller number, comparing 25 and 12, we know that the smaller between these numbers is 12. Therefore, the answer is 12.

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Hope it helps.