Answer
The solution is $\left(\frac{1}{2}, \frac{7}{2}\right)$.
Work Step by Step
We will use substitution to solve this system of equations. We substitute one of the expressions given for the $y$ term, which would mean that we are going to set the two equations equal to one another to solve for $x$ first:
$-4x^2 + 7x + 1 = 3x + 2$
We want to move all terms to the left side of the equation:
$-4x^2 + 7x - 3x + 1 - 2 = 0$
Combine like terms:
$-4x^2 + 4x - 1 = 0$
Multiply both sides of the equation by $-1$ so that the $x^2$ term is positive:
$4x^2 - 4x + 1 = 0$
Factor the quadratic equation. The quadratic equation takes the form $ax^2 + bx + c = 0$. We need to find factors whose product is $ac$ but that sum up to $b$. In this exercise, $ac = 4$ and $b = -4$. The factors $-2$ and $-2$ will work. Let's split the middle term using these factors:
$4x^2 - 2x - 2x + 1 = 0$
Group the first two terms and the last two terms:
$(4x^2 - 2x) - (2x - 1) = 0$
Factor out any common terms:
$2x(2x - 1) - (2x - 1) = 0$
Group the factors:
$(2x - 1)(2x - 1) = 0$
Set the factor equal to $0$:
First factor:
$2x - 1 = 0$
Add $1$ to each side of the equation:
$2x = 1$
Divide each side by $2$:
$x = \frac{1}{2}$
Now that we have the value for $x$, we can plug them into one of the original equations to find the corresponding $y$ value. Let's use the second equation:
$y = 3x + 2$
Substitute the solution $\frac{1}{2}$ for $x$:
$y = 3(\frac{1}{2}) + 2$
Multiply:
$y = \frac{3}{2} + 2$
Convert the constant into an equivalent fraction with $2$ as its denominator:
$y = \frac{3}{2} + \frac{4}{2}$
Add to solve:
$y = \frac{7}{2}$
The solution is $(\frac{1}{2}, \frac{7}{2})$.
