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 + 35y = 45
x = 7y - 9
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 in the first - equation

Substitute \(x = 7y-9\) into \(-5x + 35y=45\). We get \(-5(7y - 9)+35y = 45\).
Expand the left - hand side: \(-35y+45 + 35y=45\).

Step2: Simplify the equation

Combine like terms: \((-35y + 35y)+45=45\), which simplifies to \(45 = 45\).
This means the equations are dependent and the system has infinitely many solutions.
We can express \(x\) in terms of \(y\) using the second equation \(x = 7y-9\).

Answer:

The system has infinitely many solutions.
The solution set is \(\{(x,y)|x = 7y - 9\}\)