Sure! Let's break down the solution step-by-step to find the two numbers:
1. Assign Variables:
- Let [tex]\( x \)[/tex] be the smaller number.
- Let [tex]\( y \)[/tex] be the larger number.
2. Write Down the Given Information:
- We know that the sum of the two numbers is 52:
[tex]\[ x + y = 52 \][/tex]
- The larger number is three times the smaller number:
[tex]\[ y = 3x \][/tex]
3. Substitute the Second Equation into the First:
- Replace [tex]\( y \)[/tex] in the first equation with [tex]\( 3x \)[/tex]:
[tex]\[ x + 3x = 52 \][/tex]
4. Combine Like Terms:
- Combine the [tex]\( x \)[/tex] terms on the left-hand side:
[tex]\[ 4x = 52 \][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Divide both sides of the equation by 4 to isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{52}{4} \][/tex]
[tex]\[ x = 13 \][/tex]
6. Find [tex]\( y \)[/tex] Using the Value of [tex]\( x \)[/tex]:
- Now that we know [tex]\( x = 13 \)[/tex], we can substitute [tex]\( x \)[/tex] back into the equation [tex]\( y = 3x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[ y = 3 \times 13 \][/tex]
[tex]\[ y = 39 \][/tex]
7. Conclusion:
- The smaller number is [tex]\( x = 13 \)[/tex].
- The larger number is [tex]\( y = 39 \)[/tex].
So, the two numbers are 13 and 39.
