✏️NUMBERS
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[tex] \underline{\mathbb{PROBLEM}:} [/tex]
- The larger of two numbers is 5 more than twice the smaller. If the smaller is subtracted from the larger, the result is 12. What are the numbers?
[tex]\red{••••••••••••••••••••••••••••••••••••••••••••••••••} [/tex]
[tex] \underline{\mathbb{ANSWER}:} [/tex]
[tex] \qquad\LARGE » \tt\: \green{19 \: and \: 7} [/tex]
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[tex] \underline{\mathbb{SOLUTION}:} [/tex]
» Represent [tex] x [/tex] and [tex] y [/tex] as the larger and smaller number respectively. Formulate equations on the given statements.
[tex] \large \red{Eq. \: 1} [/tex] The larger of two numbers is 5 more than twice the smaller.
[tex] \large \red{Eq. \: 2} [/tex] If the smaller is subtracted from the larger, the result is 12.
- [tex] \begin{cases} x = 2y + 5 \\ x - y = 12 \end{cases} \: \begin{align} \red{(eq. \: 1)} \\ \red{(eq. \: 2)} \end{align} [/tex]
» Substitute [tex] x [/tex] from the first equation to the second equation in terms of [tex] y. [/tex]
- [tex] \begin{cases} x = 2y + 5 \\ 2y + 5 - y = 12 \end{cases} [/tex]
- [tex] \begin{cases} x = 2y + 5 \\ y + 5 = 12 \end{cases} [/tex]
- [tex] \begin{cases} x = 2y + 5 \\ y = 12 - 5 \end{cases} [/tex]
- [tex] \begin{cases} x = 2y + 5 \\ y = 7 \end{cases} [/tex]
» Thus, the smaller number is 7. Substitute it to the first equation to find [tex] x. [/tex]
- [tex] \begin{cases} x = 2(7) + 5 \\ y = 7 \end{cases} [/tex]
- [tex] \begin{cases} x = 14 + 5 \\ y = 7 \end{cases} [/tex]
- [tex] \begin{cases} x = 19 \\ y = 7 \end{cases} [/tex]
[tex] \therefore [/tex] The two numbers are 19 and 7[tex]\red{••••••••••••••••••••••••••••••••••••••••••••••••••} [/tex]
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