Certainly! Let's analyze each system of equations step-by-step to determine the number of solutions.
### System 1: [tex]\( x = y - 3 \)[/tex] and [tex]\( 2x - 2y = -6 \)[/tex]
1. Substitute [tex]\( x = y - 3 \)[/tex] into [tex]\( 2x - 2y = -6 \)[/tex]:
[tex]\[
2(y - 3) - 2y = -6
\][/tex]
Simplify:
[tex]\[
2y - 6 - 2y = -6
\][/tex]
[tex]\[
-6 = -6
\][/tex]
The equation simplifies to a true statement [tex]\((-6 = -6\)[/tex]), which means there are infinitely many solutions. The two equations are dependent, describing the same line.
Result: Infinite number of solutions
### System 2: [tex]\( 5x + 2y = -7 \)[/tex] and [tex]\( 15x + 6y = 21 \)[/tex]
1. Notice that the second equation is a multiple of the first equation:
[tex]\[
15x + 6y = 21 \quad \text{(Triple the first equation} \rightarrow 3(5x + 2y) = 3(-7) = -21\text{)}
\][/tex]
[tex]\[
15x + 6y \neq -21
\][/tex]
The second equation simplifies to [tex]\( 15x + 6y = -21 \)[/tex], indicating a contradiction [tex]\( (21 \neq -21) \)[/tex].
Result: No solution
### System 3: [tex]\( y = 2x + 1 \)[/tex] and [tex]\( 2x - y = 1 \)[/tex]
1. Substitute [tex]\( y = 2x + 1 \)[/tex] into [tex]\( 2x - y = 1 \)[/tex]:
[tex]\[
2x - (2x + 1) = 1
\][/tex]
Simplify:
[tex]\[
2x - 2x - 1 = 1
\][/tex]
[tex]\[
-1 = 1
\][/tex]
Since [tex]\( -1 \neq 1 \)[/tex], we see a contradiction, implying that no values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] satisfy both equations simultaneously.
Result: No solution
### System 4: [tex]\( 6x + 4y = 10 \)[/tex] and [tex]\( 3x + 2y = 5 \)[/tex]
1. Notice that the second equation is a multiple of the first equation:
[tex]\[
6x + 4y = 10 \quad ( \text{Double the second equation} \rightarrow 2(3x + 2y) = 2(5) = 10)
\][/tex]
[tex]\[
6x + 4y = 10
\][/tex]
Both equations represent the same line, which means every point on this line is a solution.
Result: Infinite number of solutions
In summary:
- System 1: Infinite number of solutions
- System 2: No solution
- System 3: No solution
- System 4: Infinite number of solutions
