1
algebra

How many solutions does this system have? {y=34x+23x+4y=8\begin{cases} y = -\frac{3}{4}x + 2 \\ 3x + 4y = 8 \end{cases}

A

No solution

B

Exactly one solution

C

Infinitely many solutions

D

Cannot be determined

Correct Answer: C

Choice C is the correct answer. Let's rearrange the first equation to compare with the second.

Step 1: Rearrange y=34x+2y = -\frac{3}{4}x + 2 to standard form: y+34x=2y + \frac{3}{4}x = 2

Multiply by 4: 4y+3x=84y + 3x = 8

Or rearranged: 3x+4y=83x + 4y = 8

This is exactly the second equation!

Conclusion: Infinitely many solutions (same line)

💡 Strategic Tip: Equations can look different in slope-intercept vs. standard form but represent the same line.

Choice A is incorrect because no solution requires parallel lines with different constants.

Choice B is incorrect because one solution requires different slopes.

Choice D is incorrect because we can always determine this by converting forms.