Answer:
C) Infinitely many solutions
Step-by-step explanation:
Given the systems of linear equations in standard form:
Equation 1: 24x - 9y = 3
Equation 2: 8x - 3y = 1
Transform both equations into slope-intercept form, y = mx + b:
Equation 1: 24x - 9y = 3
Subtract 24x from both sides:
24x - 24x - 9y = -24x + 3
-9y = -24x + 3
Divide both sides by -9 to isolate y:
[tex]\mathsf{\frac{-9y}{-9}\:=\:\frac{-24x + 3}{-9} }[/tex]
[tex]\mathsf{y\:=\:\frac{8}{3}x\:-\frac{1}{3}}[/tex] ⇒ This is the slope-intercept form of Equation 1, 24x - 9y = 3.
Equation 2: 8x - 3y = 1
Subtract 8x from both sides:
8x - 8x - 3y = - 8x + 1
-3y = -8x + 1
Divide both sides by -3 to isolate y:
[tex]\mathsf{\frac{-3y}{-3}\:=\:\frac{-8x + 1}{-3} }[/tex]
[tex]\mathsf{y\:=\:\frac{8}{3}x\:-\frac{1}{3}}[/tex] ⇒ This is the slope-intercept form of Equation 2, 8x - 3y = 1.
It turns out that both equations in the given system are equivalent, and their graphs will coincide on top of one another. In this case, the equations are said to be dependent, both equations will have the same solutions.
Hence, there are infinitely many solutions to the given systems of linear equations, making Option C the correct answer.
