At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine the number of solutions for each system of linear equations, we need to analyze the relationships between the equations. Here’s a step-by-step process:
1. No solution: This occurs when the lines are parallel but have different y-intercepts, meaning they never intersect. Such equations typically have the same slopes (coefficients of [tex]\(x\)[/tex]) but different constants.
2. Infinitely many solutions: This occurs when the equations describe the same line, meaning one equation can be derived by multiplying the other by a constant. They have both the same slope and the same y-intercept.
3. One solution: This occurs when the lines intersect at exactly one point, meaning their slopes are different.
Now, let’s pair each system with the correct number of solutions:
1. System:
[tex]\[ y=-4x-5 \\ y=-4x+1 \][/tex]
These equations have the same slope ([tex]\( -4 \)[/tex]) but different y-intercepts ([tex]\(-5\)[/tex] and [tex]\(1\)[/tex]). This means the lines are parallel and never intersect.
Number of solutions: No solution
2. System:
[tex]\[ -3x + y = 7 \\ 2x - 4y = -8 \][/tex]
First, we solve for [tex]\( y \)[/tex] from the first equation:
[tex]\[ y = 3x + 7 \][/tex]
Now substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ 2x - 4(3x + 7) = -8 \\ 2x - 12x - 28 = -8 \\ -10x - 28 = -8 \\ -10x = 20 \\ x = -2 \][/tex]
Now, substitute [tex]\( x \)[/tex] back into [tex]\( y = 3x + 7 \)[/tex]:
[tex]\[ y = 3(-2) + 7 = -6 + 7 = 1 \][/tex]
Thus, we have [tex]\( x = -2 \)[/tex] and [tex]\( y = 1 \)[/tex] as the solution.
Number of solutions: One solution
3. System:
[tex]\[ 3x - y = 4 \\ 6x - 2y = 8 \][/tex]
Observe that the second equation is just the first equation multiplied by 2. This means both equations describe the same line.
Number of solutions: Infinitely many solutions
So, the pairs are:
- [tex]\( y = -4x - 5 \)[/tex] and [tex]\( y = -4x + 1 \)[/tex] → No solution
- [tex]\( -3x + y = 7 \)[/tex] and [tex]\( 2x - 4y = -8 \)[/tex] → One solution
- [tex]\( 3x - y = 4 \)[/tex] and [tex]\( 6x - 2y = 8 \)[/tex] → Infinitely many solutions
1. No solution: This occurs when the lines are parallel but have different y-intercepts, meaning they never intersect. Such equations typically have the same slopes (coefficients of [tex]\(x\)[/tex]) but different constants.
2. Infinitely many solutions: This occurs when the equations describe the same line, meaning one equation can be derived by multiplying the other by a constant. They have both the same slope and the same y-intercept.
3. One solution: This occurs when the lines intersect at exactly one point, meaning their slopes are different.
Now, let’s pair each system with the correct number of solutions:
1. System:
[tex]\[ y=-4x-5 \\ y=-4x+1 \][/tex]
These equations have the same slope ([tex]\( -4 \)[/tex]) but different y-intercepts ([tex]\(-5\)[/tex] and [tex]\(1\)[/tex]). This means the lines are parallel and never intersect.
Number of solutions: No solution
2. System:
[tex]\[ -3x + y = 7 \\ 2x - 4y = -8 \][/tex]
First, we solve for [tex]\( y \)[/tex] from the first equation:
[tex]\[ y = 3x + 7 \][/tex]
Now substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ 2x - 4(3x + 7) = -8 \\ 2x - 12x - 28 = -8 \\ -10x - 28 = -8 \\ -10x = 20 \\ x = -2 \][/tex]
Now, substitute [tex]\( x \)[/tex] back into [tex]\( y = 3x + 7 \)[/tex]:
[tex]\[ y = 3(-2) + 7 = -6 + 7 = 1 \][/tex]
Thus, we have [tex]\( x = -2 \)[/tex] and [tex]\( y = 1 \)[/tex] as the solution.
Number of solutions: One solution
3. System:
[tex]\[ 3x - y = 4 \\ 6x - 2y = 8 \][/tex]
Observe that the second equation is just the first equation multiplied by 2. This means both equations describe the same line.
Number of solutions: Infinitely many solutions
So, the pairs are:
- [tex]\( y = -4x - 5 \)[/tex] and [tex]\( y = -4x + 1 \)[/tex] → No solution
- [tex]\( -3x + y = 7 \)[/tex] and [tex]\( 2x - 4y = -8 \)[/tex] → One solution
- [tex]\( 3x - y = 4 \)[/tex] and [tex]\( 6x - 2y = 8 \)[/tex] → Infinitely many solutions
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.