Question

Without graphing the equations, decide whether the system has one solution, no solution, or infinitely many solutions. 5y =x-6 4x-10y=12

Answer 1

One solution:

`\boxed{x = 0,\;y = -(6)/(5)}`

Explanation:

The solution strategy is to isolate one of the variables by eliminating the other. There are several ways to do this. This is my preferred way

The equations are

`5y = x-6 \cdots[1]\\\\4x - 10y = 12 \cdots[2]\\\\`

Transform equation [1] into the same form as [2] with x and y variables on the left and the constant on the right:

Subtract x from both sides of equation [1]:


`-x + 5y = -6\cdots[3]`

Multiply equation [3] by 2 to get the coefficients of y the same
[3] x 2 gives

`-2x + 10 y = -12\cdots[4]`

In equations [2] and [4], the coefficients of y are the same but of opposite sign

Add equations [2] and [4] to get

`4x - 10y + (-2x + 10y) = 12 + (-12)\\\\4x - 10y -2x - 10y = 12 -12\\\\4x - 2x -10y + 10y = 0\\\\2x = 0\\\\x = 0\\\\`

Substitute for x = 0 in any of the equations [1] or [2] and solve for y
Let's pick equation 1

Substitute for x = 0 in
`5y = x - 6\\`

`5y = 0 - 6\\\\5y = -6\\\\y = -(6)/(5)\\\\`

So this is a system of equations with one solution:

`\boxed{x = 0,\;y = -(6)/(5)}`