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.
x = 0.5y - 4.5
2y = 0.6x + 11.2
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)|}.
the system is inconsistent.
the equations are dependent.

Explanation:

Step1: Substitute x into second - equation

Given $x = 0.5y-4.5$, substitute it into $2y=0.6x + 11.2$. We get $2y=0.6(0.5y - 4.5)+11.2$.

Step2: Expand the right - hand side

$2y=0.6\times0.5y-0.6\times4.5 + 11.2$, which simplifies to $2y = 0.3y-2.7+11.2$.

Step3: Combine like terms

Subtract $0.3y$ from both sides: $2y-0.3y=0.3y - 0.3y-2.7 + 11.2$. So, $1.7y=8.5$.

Step4: Solve for y

Divide both sides by $1.7$: $y=\frac{8.5}{1.7}=5$.

Step5: Solve for x

Substitute $y = 5$ into $x = 0.5y-4.5$. Then $x=0.5\times5-4.5=2.5 - 4.5=-2$.

Answer:

The system has one solution. The solution set is $\{(-2,5)\}$