Question

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&15 x-5 y=-20\\\&-3 x+y=4\end{aligned} $$

Short answer

The solution to the system of equations is \(x = 0\) and \(y = 4\). The system has one unique solution.

Step-by-step solution

Identify the System Equations

The system of equations is given as: \[15x - 5y = -20\] and \[-3x + y = 4\].

Simplify the Equations

It's best to simplify the equations before proceeding. For the first equation divide by 5, yielding: \[3x - y = -4\]. The second equation remains the same.

Solve Using Substitution

Replace \(y\) in the first equation with \(-3x + 4\) (from the second equation) to get: \[3x - (-3x + 4) = -4\] which simplifies to: \[6x - 4 = -4\]. Solving this gives \(x = 0\), substituting \(x = 0\) into the second equation gives \(y = 4\).

Test for the Number of Solutions

Substitute both \(x = 0\) and \(y = 4\) into the original equations to verify if it holds true. If it does, then the system has one unique solution. If not, then it implies no solution or infinitely many solutions exist.