# How do you determine whether a linear system has one solution, many solutions, or no solution when given 5x+ 4y= -18 and 2x+3y=-24?

These have one solution

One can answer this in several ways!

Method 1

We can try directly solving the system of equations

Method 2

A bit more sophisticated method involves the coefficient matrix

The determinant of this matrix is

Method 3

The two equations can be represented graphically by two straight lines, and the question of whether solutions exist then boils down to whether the lines intersect.

Note

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You can determine the number of solutions of a linear system by analyzing the coefficients of the variables and the constants in the equations. If the system has the same number of equations as variables and the determinant of the coefficient matrix is non-zero, it has a unique solution. If the determinant is zero and the system is consistent (all equations are not contradictory), it has infinitely many solutions. If the determinant is zero and the system is inconsistent (at least one equation is contradictory), it has no solution. For the given system, the determinant is non-zero, so it has one unique solution.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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