Question

Solve the system of linear equations if possible. Does the system have exactly one solution, no solution, or infinitely many solutions? $$ \begin{aligned}&10 x-6 y=-5\\\&3 y=5 x+2\end{aligned} $$

Short answer

The solution to the system of equations is \(x = -9/20, y=1/4\). The system has exactly one solution

Step-by-step solution

Convert the second equation

First, convert the second equation from \(3y = 5x + 2\) into a similar form as the first one by subtracting \(5x\) from both sides. This gives us \(-5x + 3y = 2\)

Equate the coefficients

To solve this system of equations, we can match the coefficients in both equations. Multiply the second equation by 2 to match the coefficients with the first equation. This gives us: \(-10x + 6y = 4\)

Subtract the equations

Subtract the second equation from the first one. That is \((10x - 6y) - (-10x + 6y) = -5 - 4\). Simplification gives \(20x = -9\)

Solve for x

Divide both sides of the equation by 20 gives us \(x = -9/20\)

Solve for y

Substitute x into the second equation gives us \(3y = 5(-9/20) + 2\), which solves to give \( y = 1/4\)

Number of solutions

Since we obtained unique solutions for x and y, this system has exactly one solution