We are tasked with finding a pair of integers [tex]\(x\)[/tex] and [tex]\(y\)[/tex] such that their product is 10 and their difference is 3. Let's denote these conditions mathematically:
1. [tex]\( x \cdot y = 10 \)[/tex]
2. [tex]\( x - y = 3 \)[/tex]
To solve this, we can use a system of equations approach.
### Step-by-Step Solution
1. Represent one variable in terms of the other using the second equation.
From the second equation:
[tex]\[
x - y = 3 \implies x = y + 3
\][/tex]
Now, we have [tex]\( x \)[/tex] expressed in terms of [tex]\( y \)[/tex].
2. Substitute [tex]\( x \)[/tex] in the first equation.
Substitute [tex]\( x = y + 3 \)[/tex] into [tex]\( x \cdot y = 10 \)[/tex]:
[tex]\[
(y + 3) \cdot y = 10
\][/tex]
Simplify to form a quadratic equation:
[tex]\[
y^2 + 3y = 10
\][/tex]
[tex]\[
y^2 + 3y - 10 = 0
\][/tex]
3. Solve the quadratic equation.
The standard form of a quadratic equation is [tex]\( ay^2 + by + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -10 \)[/tex]. The discriminant ([tex]\( \Delta \)[/tex]) is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the values:
[tex]\[
\Delta = 3^2 - 4 \cdot 1 \cdot (-10) = 9 + 40 = 49
\][/tex]
4. Find the roots of the quadratic equation using the quadratic formula.
The quadratic formula is:
[tex]\[
y = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( \Delta = 49 \)[/tex]:
[tex]\[
y = \frac{-3 \pm \sqrt{49}}{2 \cdot 1} = \frac{-3 \pm 7}{2}
\][/tex]
This gives us two solutions:
[tex]\[
y_1 = \frac{-3 + 7}{2} = \frac{4}{2} = 2
\][/tex]
[tex]\[
y_2 = \frac{-3 - 7}{2} = \frac{-10}{2} = -5
\][/tex]
5. Find the corresponding [tex]\( x \)[/tex] values.
Using [tex]\( x = y + 3 \)[/tex]:
- When [tex]\( y = 2 \)[/tex]:
[tex]\[
x = 2 + 3 = 5
\][/tex]
- When [tex]\( y = -5 \)[/tex]:
[tex]\[
x = -5 + 3 = -2
\][/tex]
### Solutions
The pairs of integers that satisfy both conditions are:
1. [tex]\( (x, y) = (5, 2) \)[/tex]
2. [tex]\( (x, y) = (-2, -5) \)[/tex]
Thus, the pairs of integers whose product is 10 and whose difference is 3 are:
[tex]\[ (5, 2) \text{ and } (-2, -5) \][/tex]
