Question

identify whether the system of equations given in the first column are inconsistent, consistent and dependent, or consistent and independent.
system of equations \t consistent and independent \t inconsistent \t consistent and dependent
$2y = 4 - 6x$
$5y = -15x - 7$ \t $circ$ \t $circ$ \t $circ$
$3x + 7y = 2$
$x = \frac{2}{3} - \frac{7}{3}y$ \t $circ$ \t $circ$ \t $circ$
$30x - 6y = 3$
$5y - 10x = 4$ \t $circ$ \t $circ$ \t $circ$

Explanation:

Step1: Analyze first system

Rewrite equations to slope-intercept form ($y=mx+b$):

1. $2y=4-6x \implies y=-3x+2$
2. $5y=-15x-7 \implies y=-3x-\frac{7}{5}$

Same slope ($-3$), different y-intercepts: parallel, no solution.

Step2: Analyze second system

Rewrite second equation to standard form:

1. $3x+7y=2$
2. $x=\frac{2}{3}-\frac{7}{3}y \implies 3x+7y=2$

Equations are identical: infinitely many solutions.

Step3: Analyze third system

Rewrite equations to slope-intercept form:

1. $30x-6y=3 \implies y=5x-\frac{1}{2}$
2. $5y-10x=4 \implies y=2x+\frac{4}{5}$

Different slopes: one unique solution.

Answer:

1. System 1 ($2y=4-6x$; $5y=-15x-7$): Inconsistent
2. System 2 ($3x+7y=2$; $x=\frac{2}{3}-\frac{7}{3}y$): Consistent and Dependent
3. System 3 ($30x-6y=3$; $5y-10x=4$): Consistent and Independent