To solve the system of equations:
- ( 2x - 3y = 7 ) (Equation 1)
- ( 15x + 3y = 10 ) (Equation 2)
We can use the elimination method or substitution method. Here, let's use the elimination method.
First, we can add both equations directly, but first, let's manipulate them a little to eliminate ( y ).
From Equation 1, we can express ( y ) in terms of ( x ):
[
3y = 2x - 7 \quad \Rightarrow \quad y = \frac{2x - 7}{3}
]
Now we can plug this expression for ( y ) into Equation 2:
[
15x + 3\left(\frac{2x - 7}{3}\right) = 10
]
This simplifies to:
[
15x + (2x - 7) = 10
]
[
17x - 7 = 10
]
Now, add 7 to both sides:
[
17x = 17
]
Now, divide both sides by 17:
[
x = 1
]
Now that we know ( x ), we can substitute it back into the equation for ( y ):
[
y = \frac{2(1) - 7}{3} = \frac{2 - 7}{3} = \frac{-5}{3}
]
So the solution to the system of equations is:
[
x = 1, \quad y = -\frac{5}{3}
]
Thus, the solution is ( (x, y) = (1, -\frac{5}{3}) ).
