Question

determine whether the system of equations is consistent or inconsistent and if it is independent or dependent
2y + 10 = -5x
10x + 4y = -20
a) consistent and dependent
b) consistent and independent
c) inconsistent and dependent
d) inconsistent and independent

Explanation:

Step1: Rewrite equations in slope - intercept form

First, take the equation \(2y + 10=-5x\). Solve for \(y\):
Subtract 10 from both sides: \(2y=-5x - 10\).
Divide both sides by 2: \(y =-\frac{5}{2}x-5\).

Second, take the equation \(10x + 4y=-20\). Solve for \(y\):
Subtract \(10x\) from both sides: \(4y=-10x - 20\).
Divide both sides by 4: \(y=-\frac{10}{4}x - 5=-\frac{5}{2}x-5\).

Step2: Analyze consistency and dependence

A system of linear equations is consistent if it has at least one solution, and dependent if the two equations represent the same line (infinitely many solutions). Since both equations, when rewritten in slope - intercept form, are \(y =-\frac{5}{2}x - 5\), they represent the same line. So the system has infinitely many solutions, which means it is consistent and dependent.

Answer:

A) consistent and dependent