Question

Can you guys please help without guessing?

Answer 1

In order of appearance in the image:

Two solutions

No solutions

Infinite solutions

No solutions

Explanation:

Let's evaluate the systems of equations one by one:

`\left \{ {{x-y=3} \atop {3x+y=1}} \right.`

Let's solve this system of equations:

`x=3+y\\3(3+y)+y=1\\9+4y=1\\4y=-8\\y=-2\\x=3-2\\x=1`

This system has two solutions! x = 1, y = -2

`\left \{ {{y=5x-4} \atop {y=5x+13}} \right.`

This system of equations has no solutions because no x value exists that satisfies both equations.

`\left \{ {{4g+r=12} \atop {8g=-2r+24}} \right. \\r=12-4g\\8g=-2(12-4g)+24\\8g=-24+8g+24\\8g=8g\\1=1\\`

This system of equations has infinite solutions. No matter what values you plug in for g and r, the solution is the same!

`\left \{ {{3y+15x=-1} \atop {y=4-5x}} \right. \\3(4-5x)+15x=-1\\12-15x+15x=-1\\12=-1`

This system of equations has no solutions because no values exist that can satisfy both equations.