Sure, let's break down and solve this problem step-by-step.
Let's denote the smaller number by [tex]\( x \)[/tex] and the larger number by [tex]\( y \)[/tex].
### Step 1: Establish the equations from the given conditions
Condition 1: The larger number is 2 more than 3 times the smaller number.
This can be expressed as:
[tex]\[ y = 3x + 2 \][/tex]
Condition 2: If 3 is added to the smaller number and 1 to the larger number, they will be in the ratio 3:7.
This can be expressed as:
[tex]\[ \frac{x + 3}{y + 1} = \frac{3}{7} \][/tex]
### Step 2: Simplify the second equation
Cross-multiply to get rid of the fraction:
[tex]\[ 7(x + 3) = 3(y + 1) \][/tex]
Distribute and simplify:
[tex]\[ 7x + 21 = 3y + 3 \][/tex]
[tex]\[ 7x + 21 - 3 = 3y \][/tex]
[tex]\[ 7x + 18 = 3y \][/tex]
### Step 3: Substitute the first equation into the simplified second equation
From the first equation, we know [tex]\( y = 3x + 2 \)[/tex]. Substitute [tex]\( y \)[/tex] in the simplified second equation:
[tex]\[ 7x + 18 = 3(3x + 2) \][/tex]
[tex]\[ 7x + 18 = 9x + 6 \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Rearrange the equation to isolate [tex]\( x \)[/tex]:
[tex]\[ 7x + 18 - 9x = 6 \][/tex]
[tex]\[ -2x + 18 = 6 \][/tex]
[tex]\[ -2x = 6 - 18 \][/tex]
[tex]\[ -2x = -12 \][/tex]
[tex]\[ x = 6 \][/tex]
So, the smaller number [tex]\( x \)[/tex] is 6.
### Step 5: Find [tex]\( y \)[/tex] using the value of [tex]\( x \)[/tex]
Substitute [tex]\( x = 6 \)[/tex] into the first equation [tex]\( y = 3x + 2 \)[/tex]:
[tex]\[ y = 3(6) + 2 \][/tex]
[tex]\[ y = 18 + 2 \][/tex]
[tex]\[ y = 20 \][/tex]
So, the larger number [tex]\( y \)[/tex] is 20.
### Conclusion
The two numbers are:
- The smaller number is 6.
- The larger number is 20.
