Question

How many solutions does this system have?
2 x +3y = -3
3x + 5 = -9

Answer 1

x = 12

y = -9

The system has one solution as it has one point of intersection at (12, -9)

Explanation:

Equation 1:
`2x + 3y = -3`

Equation 2:
`3x + 5y = -9`

Rewrite equation 1 to make
`x` the subject:

`2x + 3y = -3`

`\implies 2x = -3 - 3y`

`\implies x = -\frac32 - \frac32 y`

Substitute this into equation 2 and solve for
`y`:

`3( -\frac32 - \frac32 y)+5y=-9`

`\implies -\frac92 - \frac92 y+5y=-9`

`\implies -\frac92 +\frac12 y=-9`

`\implies \frac12 y=-\frac92`

`\implies y=-9`

Substitute y = -9 into one of the equations to find
`x`:

`2x + 3(-9) = -3`

`\implies 2x - 27 = -3`

`\implies 2x = 24`

`\implies x=12`

Answer 2

x = 12 and y = -9

Explanation:

Given:

2 x +3y = -3

3x + 5y = -9

make x the subject:

`2 x +3y = -3`

`2x = -3 - 3y`

`x = (-3 - 3y)/(2)` _____equation 1

`3x + 5y = -9`

`3x = -9-5y`

`x = (-9-5y)/(3)` ______equation 2

solve them simultaneously:

`(-9-5y)/(3) = (-3 - 3y)/(2)`

`2(-9-5y) = 3(-3-3y)`

`-18-10y = -9 - 9y`

`-10y + 9y = -9 + 18`

`-y = 9`

`y = -9`

now solve for x:

`2 x +3y = -3`

`2x + 3(-9) = -3`

`2x -27 = -3`

`2x = -3 + 27`

`2x = 24`

`x = 12`

coordinates where they both intersect is (12,-9) at one point so they have one solution