Question

does the system below have a unique solution, no solution, or many solutions? what does this mean graphically?
2x + 9y = 8
x + 4.5y = 4
does the system have a unique solution, no solution, or many solutions?
a. no solution
b. a unique solution
c. many solutions
what does this mean graphically?
a. the two lines intersect at a single point.
b. the two lines are the same.
c. the two lines are parallel.

Explanation:

Step1: Rewrite equations in slope - intercept form

For the first equation $2x + 9y=8$, solve for $y$: $y=-\frac{2}{9}x+\frac{8}{9}$. For the second equation $x + 4.5y=4$, multiply through by 2 to get $2x+9y = 8$, then solve for $y$: $y=-\frac{2}{9}x+\frac{8}{9}$. Since the two equations are equivalent.

Step2: Determine number of solutions

Since the two equations represent the same line, there are infinitely many solutions (any point on the line is a solution).

Step3: Interpret graphically

Graphically, two equivalent linear equations mean the two lines are the same.

Answer:

C. many solutions
B. The two lines are the same