Question

determine whether the system has no solution, one solution, or infinitely many solutions. if the system has one solution, name it.
$-x + y = -5$
$3x + 5y = 15$
100 characters remaining

Explanation:

Step1: Solve the first equation for y

From \(-x + y = -5\), we can express \(y\) in terms of \(x\) by adding \(x\) to both sides. So, \(y = x - 5\).

Step2: Substitute y into the second equation

Substitute \(y = x - 5\) into \(3x + 5y = 15\). We get \(3x + 5(x - 5) = 15\).

Step3: Simplify and solve for x

First, expand the left - hand side: \(3x+5x - 25 = 15\). Combine like terms: \(8x-25 = 15\). Add 25 to both sides: \(8x=15 + 25=40\). Then divide both sides by 8: \(x=\frac{40}{8} = 5\).

Step4: Find the value of y

Substitute \(x = 5\) into \(y=x - 5\). Then \(y=5 - 5=0\).

Since we have found a unique pair \((x,y)=(5,0)\) that satisfies both equations, the system has one solution.

Answer:

The system has one solution, and the solution is \(x = 5,y = 0\) (or the ordered pair \((5,0)\)).