Question

3x = 2y + 5
6x - 4y = 10

Explanation:

Step1: Analyze the first equation

We have the equation \(3x = 2y + 5\). Let's rewrite it in standard form \(Ax+By = C\). Subtract \(2y\) and \(5\) from both sides: \(3x-2y=5\). Now, multiply this equation by \(2\) to see the relationship with the second equation. Multiplying each term by \(2\): \(2\times(3x - 2y)=2\times5\), which gives \(6x-4y = 10\).

Step2: Compare with the second equation

The second equation is \(6x - 4y=10\), which is exactly the same as the equation we obtained by transforming the first equation. This means the two equations are dependent (they represent the same line), so there are infinitely many solutions. The solutions are all the ordered pairs \((x,y)\) that satisfy \(3x-2y = 5\) (or equivalently \(6x - 4y=10\)). We can express \(y\) in terms of \(x\) from \(3x=2y + 5\): \(2y=3x - 5\), so \(y=\frac{3x - 5}{2}\), where \(x\) can be any real number.

Answer:

There are infinitely many solutions. The solutions are all ordered pairs \((x,y)\) such that \(y=\frac{3x - 5}{2}\) (or equivalently satisfying \(3x = 2y+5\) or \(6x - 4y = 10\)) for any real number \(x\).