To determine the solution set for the given system of equations:
1. [tex]\( x - y = 5 \)[/tex]
2. [tex]\( 2x + y = 1 \)[/tex]
we will solve this system step-by-step.
### Step 1: Solve the first equation for one of the variables
We solve the first equation for [tex]\( x \)[/tex]:
[tex]\[ x - y = 5 \][/tex]
[tex]\[ x = y + 5 \][/tex]
### Step 2: Substitute the expression for [tex]\( x \)[/tex] into the second equation
Substitute [tex]\( x = y + 5 \)[/tex] into the second equation:
[tex]\[ 2(y + 5) + y = 1 \][/tex]
### Step 3: Simplify the equation
Distribute and combine the terms:
[tex]\[ 2y + 10 + y = 1 \][/tex]
[tex]\[ 3y + 10 = 1 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ 3y = 1 - 10 \][/tex]
[tex]\[ 3y = -9 \][/tex]
[tex]\[ y = -3 \][/tex]
### Step 5: Substitute [tex]\( y \)[/tex] back into the equation from Step 1
Now that we have [tex]\( y \)[/tex], substitute [tex]\( y = -3 \)[/tex] back into [tex]\( x = y + 5 \)[/tex]:
[tex]\[ x = -3 + 5 \][/tex]
[tex]\[ x = 2 \][/tex]
### Step 6: Verify the solution
We found [tex]\( x = 2 \)[/tex] and [tex]\( y = -3 \)[/tex]. Substitute these values back into both original equations to verify:
For the first equation [tex]\( x - y = 5 \)[/tex]:
[tex]\[ 2 - (-3) = 2 + 3 = 5 \][/tex] which is true.
For the second equation [tex]\( 2x + y = 1 \)[/tex]:
[tex]\[ 2(2) + (-3) = 4 - 3 = 1 \][/tex] which is true.
### Conclusion
The solution [tex]\( x = 2 \)[/tex] and [tex]\( y = -3 \)[/tex] satisfies both equations. Therefore, the system of equations:
1. [tex]\( x - y = 5 \)[/tex]
2. [tex]\( 2x + y = 1 \)[/tex]
has exactly one solution at the point [tex]\( (2, -3) \)[/tex].
### Summary
The solution set contains one point, not the empty set or an infinite number of solutions. Therefore, the correct answer is:
1 solution.
