Write a system of equations, three equations/three unknowns, that has (1, 2, 3) as the
solution.
for this can I just write any random coefficient and then put that as equal to 1,2 or 3? or if not what type of math would I use?
yup, your first suggestion is correct
just make up something like
3x - 2y + 5z = ....
3(1) - 2(2) + 5(3) = 3-4+15 = 14
so 3x - 2y + 5z = 14 could be one of your 3 equations
just make up 2 more like that
To create a system of equations with the solution (1, 2, 3), you need to use consistent coefficients for the variables.
Let's define three variables: x, y, and z.
The general form of a linear equation is Ax + By + Cz = D, where A, B, C, and D are constants.
To get the desired solution, substitute x = 1, y = 2, and z = 3 into three different equations. Here's an example:
1x + 2y + 3z = 14 --> Equation 1
4x - 5y + 2z = 3 --> Equation 2
2x + 3y - 2z = 8 --> Equation 3
By substituting the given values, we have:
1(1) + 2(2) + 3(3) = 14
4(1) - 5(2) + 2(3) = 3
2(1) + 3(2) - 2(3) = 8
Simplifying these equations gives the following:
1 + 4 + 9 = 14
4 - 10 + 6 = 3
2 + 6 - 6 = 8
These equations are consistent with the solution (1, 2, 3), as they satisfy the system when the values of x, y, and z are plugged in.
Keep in mind that there are infinite possibilities to construct systems of equations with a specific solution, so you can adjust the coefficients or add more equations as long as they remain consistent with the given solution.