Let's solve the given simultaneous equations step-by-step.
### Given Equations:
1. [tex]\( x \cdot y = -25 \)[/tex]
2. [tex]\( y = x + 10 \)[/tex]
### Part (a) - Show that these simultaneous equations have exactly one solution
To find the solution, we can use substitution.
1. From the second equation, solve for [tex]\( y \)[/tex]:
[tex]\[
y = x + 10
\][/tex]
2. Substitute [tex]\( y = x + 10 \)[/tex] into the first equation:
[tex]\[
x \cdot (x + 10) = -25
\][/tex]
This gives us:
[tex]\[
x^2 + 10x = -25
\][/tex]
3. Rearrange to form a standard quadratic equation:
[tex]\[
x^2 + 10x + 25 = 0
\][/tex]
4. Factor the quadratic equation:
[tex]\[
(x + 5)^2 = 0
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x + 5 = 0 \implies x = -5
\][/tex]
Now, substitute [tex]\( x = -5 \)[/tex] back into the second equation to find [tex]\( y \)[/tex]:
6. Using [tex]\( y = x + 10 \)[/tex]:
[tex]\[
y = -5 + 10 = 5
\][/tex]
Hence, the solution [tex]\( (x, y) = (-5, 5) \)[/tex] satisfies both the given equations.
### Proof of uniqueness:
Given [tex]\( (x + 5)^2 = 0 \)[/tex] in step 4, the quadratic equation provided only one solution. Therefore, the simultaneous equations have exactly one solution.
### Part (b) - Solution of these simultaneous equations
The solution of the given simultaneous equations is:
[tex]\[
(x, y) = (-5, 5)
\][/tex]
Expressing the solutions to 1 decimal place, we have:
[tex]\[
x = -5.0, \quad y = 5.0
\][/tex]
These values satisfy both equations, confirming the solution.
Thus, the solution is:
[tex]\[
(x, y) = (-5.0, 5.0)
\][/tex]
