Solve the system, If a system is inconsistent or if the equations are dependent, so indica...

Solve the system, If a system is inconsistent or if the equations are dependent, so indicate. See Examples 4–5.

EXAMPLE 4

Solve the system, if possible:

Strategy Since the coefficient of the b-term is 1, we will eliminate the variable b.

WHY It is easier to determine the number needed to multiply to both sides of the equation to force opposites when one of the coefficients is 1. Then we will have a system of two equations in a and c.

Solution

We can multiply the first equation of the system by 2 and add the resulting equation to the second equation to eliminate b:

Now add the second and third equations of the system to eliminate b again:

Equations 1 and 2 form the system

Since 7a − 2c cannot equal both −4 and −5, the system is inconsistent and has no solution.

EXAMPLE 5

Solve the system:

Strategy Since the third equation does not contain the variable z, we will work with the first and second equations to obtain another equation that does not contain z,

WHY Then we can use the addition method to solve the resulting system of two equations in x and y.

Solution

We can add the first two equations to get

Since equation 1 is the same as the third equation of the system, the equations of the system are dependent, and there will be infinitely many solutions. From a graphical perspective, the equations represent three planes that intersect in a common line, as shown in figure (b).

To write the general solution of this system, we can solve equation 1 for y to get

We can then substitute 5x − 4 for y in the first equation of the system and solve for z to get

Since we have found the values of y and z in terms of x, every solution of the system has the form (x, 5x − 4, 7x − 9), where x can be any real number. For example,