`4x+7y=24`
`6x+(21)/(2)y=g`
In the system of equations above, g is a constant. If the system has infinitely many solutions, what is the value of g?

Options

A. `16`
B. `32`
C. `36`
D. `72`

Answer

C

Explanation

Difficult: Hard
Category: Heart of Algebra/ System of Linear Equations
Strategic Advice: A system of equations that has infinitely many solutions indicates that the equations in the system actually represent the same line. This means you can manipulate one of the equations to look exactly like the other other one.
Getting to the Answer: The easier way to compare the equations is to rewrite them both in slope-intercept form `(y=mx+b)`.
`4x+7y=24 to 7y=-4x+24 to y=(-4)/(7)x+(24)/(7)`
`6x+(21)/(2)y=g to (21)/(2)y=-6x+g to y=(2)/(21)*(-6x)+(2)/(21)g`
Simplifying the x term in the second equations yields `(-4)/(7)x` (which it should since the equations represent the same line), so all that's left to do is to set the y-intercepts (the constants) equal and solve for g:
`(2)/(21)g=(24)/(7)`
`(21)/(2)*(2)/(21)g=(24)/(7)*(21)/(2)`
`g=(12times2)/(7)*(7times3)/(2)`
`g=12times3`
`g=36`
The correct answer is (C).