Consistent Systems of Equations: A Guide for Math Enthusiasts

consistent system

A system with at least one solution.

A consistent system is a system of equations that has at least one solution. In other words, a consistent system has a set of values for the variables that satisfy all the equations in the system simultaneously. Mathematically, a system of equations is said to be consistent if it has a solution. If the system does not have a solution, it is said to be inconsistent.

For example, consider the following system of equations:

2x + 3y = 7
4x – 6y = 8

To determine if this system is consistent, we can check if there exist values of x and y that simultaneously satisfy both equations. One way to do this is to solve for one of the variables in terms of the other. For example, we can solve for y in the first equation as:

y = (7 – 2x)/3

Substituting this expression for y into the second equation yields:

4x – 6[(7 – 2x)/3] = 8

Simplifying this expression gives:

4x – 14 + 4x = 24

Combining like terms yields:

8x = 38

Dividing both sides by 8 gives:

x = 4.75

Now that we know the value of x, we can substitute it back into either of the original equations to solve for y. For example, using the first equation we have:

2(4.75) + 3y = 7

Solving for y gives:

y = 0.5

So the solution for this system is (4.75, 0.5). Since we were able to find a solution, this system is consistent.

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