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
