Find all real solutions to the system of equations using the substitution method.
-

Question

Quizplus · Accepted Answer

We can solve for $y$ in the first equation:
$$y = 4 - x$$
and substitute this expression for $y$ into the second equation:
$$x-3(4-x)=-6$$
Simplifying this equation gives:
$$x-12+3x=-6$$
Solving for $x$ gives $x=3$, so we can substitute this value of $x$ back into the equation we derived for $y$ to get $y=1$. Thus, the solution is $(3,1)$. Therefore, the answer is choice C. First, solve one of the equations for one of the variables in terms of the other variable. For example, in the second equation, we can solve for $y$:
$$y = 3 - \frac{1}{x}$$
Now substitute this expression for $y$ into the first equation:
$$x\sqrt{3 - \frac{1}{x}} = 2$$
Square both sides:
$$x^2\left(3 - \frac{1}{x}\right) = 4$$
Simplify and solve for $x$:
$$3x - 1 = \frac{4}{x} \implies 3x^2 - x - 4 = 0$$
This factors as $(3x+4)(x-1)=0$, so the solutions are $x=-\frac{4}{3}$ and $x=1$. Plugging these values into the equation we derived for $y$, we get $y=4/3$ and $y=2$, respectively. Therefore, the solutions to the system of equations are $(x,y) = \left(-\frac{4}{3}, \frac{4}{3}\right)$ and $(x,y) = (1,2)$, which correspond to choice C.

Find All Real Solutions to the System of Equations Using