if a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this.


m - 2n = 16,

4m + n = 1

this one is not hard to solve,

mentally double the second equation and add that result to the first one ...
9m = 18
m = 2
mentally sub that into the second equation...
8+n=1
n = -7

To determine the solution set using set-builder notation, we need to solve the system of equations first. Let's begin by solving the given system:

Equation 1: m - 2n = 16
Equation 2: 4m + n = 1

We can solve the system of equations using the method of substitution or elimination. Let's use the method of elimination:

Multiply Equation 1 by 4 to make the coefficients of m in both equations the same:
4(m - 2n) = 4(16)
4m - 8n = 64

Now we have the system:
4m - 8n = 64
4m + n = 1

Subtracting the second equation from the first equation eliminates the m variable:
(4m - 8n) - (4m + n) = 64 - 1
4m - 8n - 4m - n = 63
-9n = 63
n = -7

Substituting the value of n back into Equation 2:
4m + (-7) = 1
4m - 7 = 1
4m = 8
m = 2

So the solution to the system is m = 2 and n = -7.

Now, let's determine if the system has an infinite number of solutions or no solution. To do this, we can check if the two equations are equivalent or parallel. If they are equivalent, then the system has an infinite number of solutions. If they are parallel and never intersect, the system has no solution.

We can rewrite the first equation in terms of m:
m = 16 + 2n

Now compare this with the second equation:
4m + n = 1

Substituting the expression for m from the first equation into the second equation:
4(16 + 2n) + n = 1
64 + 8n + n = 1
9n = -63
n = -7

Now, substitute this value of n back into the first equation:
m = 16 + 2(-7)
m = 16 - 14
m = 2

Now we can see that the values of m and n obtained from both equations match the values we previously found by solving the system. Therefore, the two equations are equivalent, and the system has infinitely many solutions.

Using set-builder notation, we can write the solution set as:
{(m, n) | m = 2, n = -7}

Therefore, the solution set to the given system of equations is {(2, -7)}.