If a linear pair of Equations is Consistent

Suppose a pair of linear equations is consistent. Then it can be proved by finding the solution set of our system and checking to see if it contains the interval (x,y) as a subset.

To find the solution set, we need to use matrices and determinants. First, we must represent our system as AX=B in the matrix.

Then look at how to find the determinant of the pair. First, we notice that for the matrix form of the equations, A=B and the determinant of both matrices is zero since it is equal to their product.

What Is a Linear Equation?

A linear equation contains variables, as opposed to constants. An example is y=4x+1 or y=2x-3. They contain a term with a variable raised to the first power, a coefficient, and another variable raised to some exponent.

Now we can use this information to prove if our system is consistent or not. If we are trying to check the consistency of a system of three equations in three unknowns, we would need three matrices, three unknowns, and three determinants.

ax+by=c….. (1)

is equivalent to: 

Ax=c …..(2)

Since A=B, we get: Bx=c ……(3)

To find the solution set of our system, we need to solve equations 2 and 3 simultaneously. Let’s multiply equation two by e on both sides to do this.

e(Bx)=e(c) …..(4)

Then, substitute equation 4 into 3 and simplify.

Checking for Consistency

The determinant of the pair (matrix form) A=B is equal to the product of their determinants.

We have to check if the solutions are all between 1 and 10. To do this, we multiply the first solution set by 10 to find all the possible values of the second solution set. This gives us our answer.

“If a pair of linear equations is consistent, then the solution set of our system has to contain all possible combinations of 3, 6, and 9.” Because the solution set of (2, 3) is {9,6,3}, it means that the equations are consistent.

“If a pair of linear equations is inconsistent, then the solution set of our system has to contain all possible combinations of 3 and 4.” Because the solution set of (2, 3) is {4}, it means that the equations are inconsistent.

If a set of linear equations is consistent, we can find all the possible solutions by first multiplying one solution by a number and then finding the product of all the solutions. If a set of linear equations is inconsistent, there are no possible solutions.

If a pair of linear equations is consistent, then the solution set of our system has to contain all possible combinations of 3, 6, and 9. To check this for sure, we need to make sure that all possible values for solving equations 4 and 5 are within the interval (1,10).

We can check whether two linear equations are consistent with a determinant. First, we notice that for the matrix form of the equations, A=B and the determinant of both matrices is zero since it is equal to their product.

How to Check the Consistency of Linear Equations Using Matrices

We can check whether two linear equations are consistent with a determinant. First, we notice that for the matrix form of the equations, A=B and the determinant of both matrices is zero since it is equal to their product if a pair of linear equations is consistent.

We have to check if the solutions are all between 1 and 10. To do this, we multiply the first solution set by 10 to find all the possible values of the second solution set. This gives us our answer.

Proof: We conclude that for the matrix form of our equations, A=B and the determinant of both matrices is zero since it is equal to their product:

We begin by finding the solution space to A=B (1) and (2).

First, multiply (1) by a to find the solution set #2. Multiply (2) by e to find the solution set #3. 

Substitute equation 4 into 2 and simplify to obtain #4. Prove that all possible solutions must be contained in intervals 1,10, which would be #5. 

Using e, multiply #4 by e, then simplify to obtain the answer. Divide by #5 to obtain the answer.

Conclusion

In a consistent pair of linear equations, an entire term with a variable raised to the first power is replaced by a constant. For example, the constant in y=3x+3 would be ignored. The content in y=2x-8 would be treated as three instead. 

The variables are treated as constants. The consistency in y=5x+9 would be treated as 5. We assume that the coefficients of the linear equation are all consistent.

Two linear equations are said to be equivalent if they are written in the same form. The same graph or equation can represent them.