Linear Systems Equation Solutions Questions

Question

Determine how many solutions each of the following linear systems have:

(a) 6x + 8y = 5

−2x + 9y = −6

(b) 5x + 2y + z = 4

−x + 8y + z = −2

(c) 2x − 3y = −1

3x + y = −4

x − 7y = 2

(d) x + 3y + 3z = 3

x − y − z = 4

2x + y + z = 6

Answer 1

a) unique solution

b) infinitely many solutions. One way to see this: pick any value of z. You can solve for x and y by Gaussian elimination.

c) unique solution. Reorder the equations so that x - 7y = 2 is first. Therefore -2x + 14y = -4.

Next equation: 11y= -5 (Add -2 * (x - 7x = 2) to 2x - 3y = -1)

Third equation: 22y = -10 (Add -3 * (x - 7x = 2) to 3x + y = -4)

The second and third equations are redundant. By inspection, the following system has a unique solution:

x - 7y = 2

11y= -5

d) no solutions. subtract (x + 3y + 3z = 3) from (x - y - z = 4) to get -4y - 4z = 1

add -2 * (x + 3y + 3z = 3) to (2x + y + z = 6) to get -5y - 5z = 0. So you have a contradiction.