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There are two distinct ways to solve these kinds of problems, for which the applicability strongly depends on your mathematical abilities. The first one is easier in 2D space and comes down to simply drawing the lines in a grid and determine which area satisfies all of the constraints. As your problem is a 3D problem, it becomes a bit difficult to picture that graphically so let's move to the second approach.
The second approach is via reduction, where you really have to pay attention to your bookkeeping. If you're comfortable with matrix calculation, you can translate your problem into matrices, which effectively does the same but looks smarter. Let's assume we stay with the ordinary representation as you included above.
You have 3 equations and 3 variables. First, do a quick check to make sure those 3 equations are 'independent', which means that you have to check if they are all different equations from a scaling point of view. 2x + 4y = 1 is in this case the same equation at 4x + 8y = 2, because you can scale the first one via multiplying by 2 into the next. That means both equations are the same and you can only solve 3 variables if you have 3 independent equations (or more, but that's another topic).
Solve the problem by isolating a selected variable (x, y or z) from two of the three equations. For instance, take the first one and solve for x ==> x = 2y - z. Do the same for the 3rd equation ==> x = -2y+5z. Now use these two new equations and combine with the one equation you have not yet touched (so 2x – 3y – 4z = –9 in this example).
This gives you only two new equations, but without X as variable. Now you have reduced your problem from 3 variables and 3 equations to 2 variable and only 2 equations.
Do the same thing again. Take one of the two remaining equations with only y and z and solve for either y or z. Combine this new equation with the remaining one and you end up with one variable and one equation, which can be solved mathemetically. This gets you a firm value for this variable. Go back to the equations with only 2 variables, use the known one and solve variable number 2. Again one step up to the 3 equations where you can fill out two known ones and calculate the single remaining one. Problem solved...I leave it up to you to actually do it, so pls share your results if you want any support in that.