Certainly! Let's solve the given problem step-by-step.
### Problem 1:
Given:
[tex]\[ a \cdot (x + y) = 13 \][/tex]
[tex]\[ x \cdot y = 22 \][/tex]
To find:
[tex]\[ x^2 + y^2 \][/tex]
Step-by-step solution:
1. From the first equation, we know:
[tex]\[ a \cdot (x + y) = 13 \][/tex]
2. To simplify our work, let [tex]\(a = 1\)[/tex] and solve it directly as:
[tex]\[ x + y = 13 \][/tex]
3. From the second equation, we have:
[tex]\[ x \cdot y = 22 \][/tex]
4. We aim to find [tex]\( x^2 + y^2 \)[/tex]. We can use the identity:
[tex]\[ x^2 + y^2 = (x + y)^2 - 2xy \][/tex]
5. Substituting the known values into the identity:
[tex]\[ x^2 + y^2 = (13)^2 - 2 \cdot 22 \][/tex]
6. Calculate the values:
[tex]\[ x^2 + y^2 = 169 - 44 \][/tex]
[tex]\[ x^2 + y^2 = 125 \][/tex]
So, the value of [tex]\( x^2 + y^2 \)[/tex] is [tex]\( 125 \)[/tex].
### Problem 2:
Given:
[tex]\[ 9x^2 + y^2 = 397 \][/tex]
[tex]\[ x \cdot y = 38 \][/tex]
To find:
[tex]\[ 3x + y \][/tex]
Step-by-step solution:
1. Given equations:
[tex]\[ 9x^2 + y^2 = 397 \][/tex]
[tex]\[ x \cdot y = 38 \][/tex]
2. Let's start by recognizing any possible symmetry or transformation in the equations without immediate need for specified values:
- We need to arrange it in a way to solve for [tex]\(3x + y\)[/tex].
3. Square [tex]\(3x + y\)[/tex] and use the given equations:
[tex]\[ (3x + y)^2 = 9x^2 + 2 \cdot 3x \cdot y + y^2 \][/tex]
4. Substitute the given values:
[tex]\[ (3x + y)^2 = 9x^2 + 6xy + y^2 \][/tex]
[tex]\[ (3x + y)^2 = 397 + 6 \cdot 38 \][/tex]
[tex]\[ (3x + y)^2 = 397 + 228 \][/tex]
[tex]\[ (3x + y)^2 = 625 \][/tex]
5. Taking the square root of both sides:
[tex]\[ 3x + y = \sqrt{625} \][/tex]
[tex]\[ 3x + y = 25 \][/tex]
So, the value of [tex]\( 3x + y \)[/tex] is [tex]\( 25 \)[/tex].
