Final answer:
The solutions are: x = 3, y = -2 and x = -3, y = 10.
Step-by-step explanation:
To solve the system of equations xy + 3x = 3 and 3x + y = 7, we can use the method of substitution.
Step 1: Solve one of the equations for one variable.
From the second equation, we can isolate y:
y = 7 - 3x
Step 2: Substitute the expression for y in the other equation.
Substituting y = 7 - 3x into the first equation gives:
xy + 3x = 3
x(7 - 3x) + 3x = 3
7x - 3x^2 + 3x = 3
-3x^2 + 10x - 3 = 0
Step 3: Solve the resulting quadratic equation.
We can use factoring, completing the square, or the quadratic formula to solve the quadratic equation. In this case, let's use the quadratic formula.
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
x = (-10 ± √(10^2 - 4(-3)(-3))) / (2(-3))
x = (-10 ± √(100 - 36)) / (-6)
x = (-10 ± √64) / (-6)
x = (-10 ± 8) / (-6)
Step 4: Calculate the values of x and y.
Plugging the values of x into the equation y = 7 - 3x, we get:
For x = (-10 + 8) / (-6), y = 7 - 3((-10 + 8) / (-6))
For x = (-10 - 8) / (-6), y = 7 - 3((-10 - 8) / (-6))
The solutions are:
x = 3, y = -2
x = -3, y = 10