Let's solve each system of equations step by step and match them with the appropriate solutions, which are [tex]\((-2, 0)\)[/tex], infinite number of solutions, or no solution.
### 1. System:
[tex]\[ 4x - 3y = -1 \][/tex]
[tex]\[ -3x + 4y = 6 \][/tex]
Multiply the first equation by 3 and the second by 4 to align the coefficients:
[tex]\[ 12x - 9y = -3 \][/tex]
[tex]\[ -12x + 16y = 24 \][/tex]
Adding these equations:
[tex]\[ 12x - 9y - 12x + 16y = -3 + 24 \][/tex]
[tex]\[ 7y = 21 \][/tex]
[tex]\[ y = 3 \][/tex]
Substituting [tex]\( y = 3 \)[/tex] back into the first equation:
[tex]\[ 4x - 3(3) = -1 \][/tex]
[tex]\[ 4x - 9 = -1 \][/tex]
[tex]\[ 4x = 8 \][/tex]
[tex]\[ x = 2 \][/tex]
Solution: [tex]\((2, 3)\)[/tex]
### 2. System:
[tex]\[ 3x - 2y = -1 \][/tex]
[tex]\[ -x + 2y = 3 \][/tex]
Add both equations:
[tex]\[ 3x - 2y - x + 2y = -1 + 3 \][/tex]
[tex]\[ 2x = 2 \][/tex]
[tex]\[ x = 1 \][/tex]
Substitute [tex]\( x = 1 \)[/tex] back into the second equation:
[tex]\[ -1 + 2y = 3 \][/tex]
[tex]\[ 2y = 4 \][/tex]
[tex]\[ y = 2 \][/tex]
Solution: [tex]\((1, 2)\)[/tex]
### 3. System:
[tex]\[ 3x + 6y = 6 \][/tex]
[tex]\[ 2x + 4y = -4 \][/tex]
Notice both equations are multiples of each other:
[tex]\[ \text{Divide the first by 3:} \][/tex]
[tex]\[ x + 2y = 2 \][/tex]
[tex]\[ \text{Divide the second by 2:} \][/tex]
[tex]\[ x + 2y = -2 \][/tex]
Contradiction as the same left-hand side equals different right-hand sides. Hence, there is no solution.
### 4. System:
[tex]\[ -3x + 6y = -3 \][/tex]
[tex]\[ 5x - 10y = 5 \][/tex]
Divide both equations by their leading coefficients:
[tex]\[ -3x + 6y = -3 \][/tex]
[tex]\[ x - 2y = -1 \][/tex] (Dividing the first equation by -3)
[tex]\[ 5x - 10y = 5 \][/tex]
[tex]\[ x - 2y = 1 \][/tex] (Dividing the second equation by 5)
Contradiction as the same left-hand side equals different right-hand sides. Hence, there are infinite solutions.
### Summary:
- [tex]\((2, 3)\)[/tex]: No match found from the provided solutions.
- [tex]\((1, 2)\)[/tex]: No match found from the provided solutions.
- The pair [tex]\(\ 4x - 3y = -1 \)[/tex][tex]\( -3 x + 4 y =6 matches: User error
- The pair \(3 x - 2 y = -1 \)[/tex] [tex]\(-x + 2 y = 3\)[/tex]
matches: User error
with [tex]\(x=-2, y=0\)[/tex]}: No match found from the provided solutions.
Notice both equations are multiples of each other:
Hence,
- The pair [tex]\(\ 3x + 6y = 6 and {2 x + 4 y =-4\)[/tex] matches Infinite number of solutions.
- The pair [tex]\( -3 x + 6 y=-3 \)[/tex] and {[tex]\(}5 x-10 y=5\)[/tex] No Solution.
