To solve a system of equations, such as 5x + 3y = 10 and -7x + 5y = 15, one effective method is to manipulate the equations so that one variable can be eliminated. This can be achieved by multiplying each equation by the coefficients of the opposite equation. In this case, the coefficients of x are 5 and -7, while the coefficients of y are 3 and 5.

First, we check if any coefficients are equal with opposite signs or if they are factors of each other. Here, 5 and -7 are not equal, nor are they factors of each other. Therefore, we can proceed by multiplying the first equation by 7 and the second equation by 5. This approach ensures that the coefficients of x will become equal in magnitude but opposite in sign:

Multiplying the first equation by 7:

7(5x + 3y) = 7(10)

Results in:

35x + 21y = 70

Multiplying the second equation by 5:

5(-7x + 5y) = 5(15)

Results in:

-35x + 25y = 75

Now, we can add these two new equations together:

(35x + 21y) + (-35x + 25y) = 70 + 75

This simplifies to:

46y = 145

To find the value of y, divide both sides by 46:

y = \frac{145}{46}

Although this value may not be a whole number, it is still valid. Next, substitute this value of y back into one of the original equations to solve for x. This method of multiplying equations to eliminate variables is a reliable strategy for solving systems of linear equations.