Question

solve the system using the substitution method. if the system does not have one unique solution, al inconsistent or the equations are dependent. express numbers in exact simplified form.
y = 1/3x - 11/3
x = 2/5y + 16/5
the system has one solution.
the solution set is {()}.
the system has no solution, {}.
the system is inconsistent.
the equations are dependent.
the system has infinitely many solutions.
the solution set is {(x,y)|}.
the system is inconsistent.
the equations are dependent.

Explanation:

Step1: Substitute the first - equation into the second

Substitute $y=\frac{1}{3}x - \frac{11}{3}$ into $x=\frac{2}{5}y+\frac{16}{5}$.
$x=\frac{2}{5}(\frac{1}{3}x-\frac{11}{3})+\frac{16}{5}$

Step2: Expand the right - hand side

$x=\frac{2}{15}x-\frac{22}{15}+\frac{16}{5}$

Step3: Get a common denominator

$x=\frac{2}{15}x-\frac{22}{15}+\frac{48}{15}$

Step4: Combine like terms

$x-\frac{2}{15}x=\frac{-22 + 48}{15}$
$\frac{15x-2x}{15}=\frac{26}{15}$
$\frac{13x}{15}=\frac{26}{15}$

Step5: Solve for x

Multiply both sides by $\frac{15}{13}$, we get $x = 2$.

Step6: Solve for y

Substitute $x = 2$ into $y=\frac{1}{3}x-\frac{11}{3}$, then $y=\frac{1}{3}\times2-\frac{11}{3}=\frac{2 - 11}{3}=-3$.

Answer:

The system has one solution. The solution set is $\{(2,-3)\}$