Question

Choose an equation that when paired with the equation below, will create a system of equations with infinitely many solutions, one solution, or no solutions.

Answer 1

Infinitely many solutions:
`-5x+9y=2`

One solution:
`-10x+5y=4`

No solution:
`-10x+18y=5`

Explanation:

Solve for y from each equation to obtain the equation of the line in slope-intercept form:

`y=mx+b`

m: slope

b: y-intercept

EQUATION ON THE TOP:

`-10x+18y=4\\\\y=(5)/(9)x+(2)/(9)`

Equation 1:

`5y=10x+4\\\\y=2x+(4)/(5)`

The equation on top and the equation equation 1 has different slopes and different y-intercept, therefore the system will have one solution.

Equation 2:

`9y=5x+2\\\\y=(5)/(9)x+(2)/(9)`

The equation on top and the equation equation 2 are the same, therefore, the system will have infinitely many solutions.

Equation 3:

`18y=10x+5\\\\y=(5)/(9)x+(5)/(18)`

The slopes are equal, then both lines will be parallels, therefpre the system will have no solution.