Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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.
### 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.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
