Question

solve the system using the substitution method. if the system does not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. express numbers in exact simplified form.
5x = 2(3y + 2)
3x - 4y = 2
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)|}

Explanation:

Step1: Simplify the first equation

First, expand $5x = 2(3y + 2)$ to get $5x=6y + 4$, then express $x$ in terms of $y$: $x=\frac{6y + 4}{5}$.

Step2: Substitute $x$ into the second - equation

Substitute $x=\frac{6y + 4}{5}$ into $3x-4y = 2$. So we have $3\times\frac{6y + 4}{5}-4y=2$.
Multiply through by 5 to clear the fraction: $3(6y + 4)-20y = 10$.
Expand the left - hand side: $18y+12 - 20y=10$.
Combine like terms: $-2y=10 - 12=-2$.
Solve for $y$: $y = 1$.

Step3: Find the value of $x$

Substitute $y = 1$ into $x=\frac{6y + 4}{5}$. Then $x=\frac{6\times1+4}{5}=\frac{10}{5}=2$.

Answer:

The system has one solution. The solution set is $\{(2,1)\}$