Question

Classify each system. the system shown below is . x + 3y = 4 3x + 9y = 12 the system shown below is . 3x – 4y = 12 6x – 8y = 21 the system shown below is . 2x – 3y = 8 –3x + 2y = 8

Answer 1

The first system is consistent & dependent; the second system is inconsistent; and the third system is consistent & independent.

Explanation:

A consistent system is a system that has at least one solution. A dependent system is a system that has infinite solutions, because the equations describe the same line. An independent system is a system that has one solution.

An inconsistent system is a system that has no solution because the lines are parallel.

The first system is

`\left \{ {{x+3y=4} \atop {3x+9y=12}} \right.`

To solve this, we would make the coefficients of y the same. We would do this by multiplying the top equation by 3:

`\left \{ {{3(x+3y=4)} \atop {3x+9y=12}} \right. \\\\\left \{ {{3x+9y=12} \atop {3x+9y=12}} \right.`

We have the same equation; thus this system is consistent and dependent.

The second system is

`\left \{ {{3x-4y=12} \atop {6x-8y=21}} \right.`

To solve this, we will make the coefficients of x the same; we will do this by multiplying the top equation by 2:

`\left \{ {{2(3x-4y=12)} \atop {6x-8y=21}} \right. \\\\\left \{ {{6x-8y=24} \atop {6x-8y=21}} \right.`

We have two equations that have the same slope but different y-intercepts; these lines are parallel and thus are inconsistent.

The third system is

`\left \{ {{2x-3y=8} \atop {-3x+2y=8}} \right.`

To solve this, we will make the coefficients of x the same; we will do this by multiplying the top equation by 3 and the bottom equation by 2:

`\left \{ {{3(2x-3y=8)} \atop {2(-3x+2y=8)}} \right. \\\\\left \{ {{6x-9y=24} \atop {-6x+4y=16}} \right.`

Now we will add the two equations:

`\left \{ {{6x-9y=24} \atop {+(-6x+4y=16)}} \right. \\\\-5y = 40`

Divide both sides by -5:

-5y/-5 = 40/-5

y = -8

Now substitute this into the first equation:

2x-3(-8) = 8

2x+24 = 8

Subtract 24 from each side:

2x+24 - 24 = 8-24

2x = -16

Divide both sides by 2:

2x/2 = -16/2

x = -8

The solution is (-8, -8); thus the system is consistent and independent.

Answer 2

The first one is consitent and dependent

The second one is inconsistent

The third one is consistent and independent