Answer :
Sure! Let's solve this system of equations step-by-step:
The given system is:
[tex]\[
\begin{array}{l}
|x + y| = 7 \\
|x - y| = 913
\end{array}
\][/tex]
Let’s consider different cases for the absolute values.
### Case 1: [tex]\( x + y = 7 \)[/tex]
In this case, the equations become:
[tex]\[
\begin{aligned}
x + y &= 7 \\
|x - y| &= 913
\end{aligned}
\][/tex]
Subcase 1a: [tex]\( x - y = 913 \)[/tex]
[tex]\[
\begin{aligned}
x + y &= 7 \\
x - y &= 913
\end{aligned}
\][/tex]
Adding these equations:
[tex]\[
\begin{aligned}
(x + y) + (x - y) &= 7 + 913 \\
2x &= 920 \\
x &= 460
\end{aligned}
\][/tex]
Substituting [tex]\( x = 460 \)[/tex] back into [tex]\( x + y = 7 \)[/tex]:
[tex]\[
\begin{aligned}
460 + y &= 7 \\
y &= 7 - 460 \\
y &= -453
\end{aligned}
\][/tex]
So, one solution is [tex]\( (x, y) = (460, -453) \)[/tex].
Subcase 1b: [tex]\( x - y = -913 \)[/tex]
[tex]\[
\begin{aligned}
x + y &= 7 \\
x - y &= -913
\end{aligned}
\][/tex]
Adding these equations:
[tex]\[
\begin{aligned}
(x + y) + (x - y) &= 7 - 913 \\
2x &= -906 \\
x &= -453
\end{aligned}
\][/tex]
Substituting [tex]\( x = -453 \)[/tex] back into [tex]\( x + y = 7 \)[/tex]:
[tex]\[
\begin{aligned}
-453 + y &= 7 \\
y &= 7 + 453 \\
y &= 460
\end{aligned}
\][/tex]
So, another solution is [tex]\( (x, y) = (-453, 460) \)[/tex].
### Case 2: [tex]\( x + y = -7 \)[/tex]
In this case, the equations become:
[tex]\[
\begin{aligned}
x + y &= -7 \\
|x - y| &= 913
\end{aligned}
\][/tex]
Subcase 2a: [tex]\( x - y = 913 \)[/tex]
[tex]\[
\begin{aligned}
x + y &= -7 \\
x - y &= 913
\end{aligned}
\][/tex]
Adding these equations:
[tex]\[
\begin{aligned}
(x + y) + (x - y) &= -7 + 913 \\
2x &= 906 \\
x &= 453
\end{aligned}
\][/tex]
Substituting [tex]\( x = 453 \)[/tex] back into [tex]\( x + y = -7 \)[/tex]:
[tex]\[
\begin{aligned}
453 + y &= -7 \\
y &= -7 - 453 \\
y &= -460
\end{aligned}
\][/tex]
So, another solution is [tex]\( (x, y) = (453, -460) \)[/tex].
Subcase 2b: [tex]\( x - y = -913 \)[/tex]
[tex]\[
\begin{aligned}
x + y &= -7 \\
x - y &= -913
\end{aligned}
\][/tex]
Adding these equations:
[tex]\[
\begin{aligned}
(x + y) + (x - y) &= -7 - 913 \\
2x &= -920 \\
x &= -460
\end{aligned}
\][/tex]
Substituting [tex]\( x = -460 \)[/tex] back into [tex]\( x + y = -7 \)[/tex]:
[tex]\[
\begin{aligned}
-460 + y &= -7 \\
y &= -7 + 460 \\
y &= 453
\end{aligned}
\][/tex]
So, the last solution is [tex]\( (x, y) = (-460, 453) \)[/tex].
### Conclusion:
The four solutions are:
1. [tex]\( (460, -453) \)[/tex]
2. [tex]\( (-453, 460) \)[/tex]
3. [tex]\( (453, -460) \)[/tex]
4. [tex]\( (-460, 453) \)[/tex]
Thus, the system has [tex]\( \boxed{four\ solutions} \)[/tex].
The given system is:
[tex]\[
\begin{array}{l}
|x + y| = 7 \\
|x - y| = 913
\end{array}
\][/tex]
Let’s consider different cases for the absolute values.
### Case 1: [tex]\( x + y = 7 \)[/tex]
In this case, the equations become:
[tex]\[
\begin{aligned}
x + y &= 7 \\
|x - y| &= 913
\end{aligned}
\][/tex]
Subcase 1a: [tex]\( x - y = 913 \)[/tex]
[tex]\[
\begin{aligned}
x + y &= 7 \\
x - y &= 913
\end{aligned}
\][/tex]
Adding these equations:
[tex]\[
\begin{aligned}
(x + y) + (x - y) &= 7 + 913 \\
2x &= 920 \\
x &= 460
\end{aligned}
\][/tex]
Substituting [tex]\( x = 460 \)[/tex] back into [tex]\( x + y = 7 \)[/tex]:
[tex]\[
\begin{aligned}
460 + y &= 7 \\
y &= 7 - 460 \\
y &= -453
\end{aligned}
\][/tex]
So, one solution is [tex]\( (x, y) = (460, -453) \)[/tex].
Subcase 1b: [tex]\( x - y = -913 \)[/tex]
[tex]\[
\begin{aligned}
x + y &= 7 \\
x - y &= -913
\end{aligned}
\][/tex]
Adding these equations:
[tex]\[
\begin{aligned}
(x + y) + (x - y) &= 7 - 913 \\
2x &= -906 \\
x &= -453
\end{aligned}
\][/tex]
Substituting [tex]\( x = -453 \)[/tex] back into [tex]\( x + y = 7 \)[/tex]:
[tex]\[
\begin{aligned}
-453 + y &= 7 \\
y &= 7 + 453 \\
y &= 460
\end{aligned}
\][/tex]
So, another solution is [tex]\( (x, y) = (-453, 460) \)[/tex].
### Case 2: [tex]\( x + y = -7 \)[/tex]
In this case, the equations become:
[tex]\[
\begin{aligned}
x + y &= -7 \\
|x - y| &= 913
\end{aligned}
\][/tex]
Subcase 2a: [tex]\( x - y = 913 \)[/tex]
[tex]\[
\begin{aligned}
x + y &= -7 \\
x - y &= 913
\end{aligned}
\][/tex]
Adding these equations:
[tex]\[
\begin{aligned}
(x + y) + (x - y) &= -7 + 913 \\
2x &= 906 \\
x &= 453
\end{aligned}
\][/tex]
Substituting [tex]\( x = 453 \)[/tex] back into [tex]\( x + y = -7 \)[/tex]:
[tex]\[
\begin{aligned}
453 + y &= -7 \\
y &= -7 - 453 \\
y &= -460
\end{aligned}
\][/tex]
So, another solution is [tex]\( (x, y) = (453, -460) \)[/tex].
Subcase 2b: [tex]\( x - y = -913 \)[/tex]
[tex]\[
\begin{aligned}
x + y &= -7 \\
x - y &= -913
\end{aligned}
\][/tex]
Adding these equations:
[tex]\[
\begin{aligned}
(x + y) + (x - y) &= -7 - 913 \\
2x &= -920 \\
x &= -460
\end{aligned}
\][/tex]
Substituting [tex]\( x = -460 \)[/tex] back into [tex]\( x + y = -7 \)[/tex]:
[tex]\[
\begin{aligned}
-460 + y &= -7 \\
y &= -7 + 460 \\
y &= 453
\end{aligned}
\][/tex]
So, the last solution is [tex]\( (x, y) = (-460, 453) \)[/tex].
### Conclusion:
The four solutions are:
1. [tex]\( (460, -453) \)[/tex]
2. [tex]\( (-453, 460) \)[/tex]
3. [tex]\( (453, -460) \)[/tex]
4. [tex]\( (-460, 453) \)[/tex]
Thus, the system has [tex]\( \boxed{four\ solutions} \)[/tex].
