Question

without graphing, determine whether the system of linear equations has one solution, infinitely many solutions, or no solution. explain your reasoning.
1.
y = -2x - 4
y = 2x - 4
2.
y = -6x - 8
y = -6x + 8
3.
3x - y = 6
-3x + y = -6

Explanation:

1. Step1: Compare slopes

Slopes are -2 and 2 (different).

1. Step2: Conclusion

Different slopes mean one intersection (one solution).

2. Step1: Compare slopes and intercepts

Slopes are both -6 (same), intercepts -8 vs 8 (different).

2. Step2: Conclusion

Same slope + different intercepts = parallel lines (no solution).

3. Step1: Rewrite second equation

$-3x + y = -6$ → $y = 3x - 6$.

3. Step2: Compare with first equation

First equation: $3x - y = 6$ → $y = 3x - 6$. Identical equations.

3. Step3: Conclusion

Identical lines → infinitely many solutions.

Answer:

1. One solution
2. No solution
3. Infinitely many solutions