$y=-3 x$ $\newline$ $\frac{y-7}{6}=\frac{3 x^{2}+x-5}{2}$ $\newline$ If $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ are the distinct solutions to the system of equations shown, what is the sum of the $y_{1}+y_{2}$ ?

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Algebra 1

Compare linear and exponential growth

Full solution

y=-3 x

\newline

\frac{y-7}{6}=\frac{3 x^{2}+x-5}{2}

\newline

\left(x_{1}, y_{1}\right)

and

\left(x_{2}, y_{2}\right)

are the distinct solutions to the system of equations shown, what is the sum of the

y_{1}+y_{2}

Write Equations: Write down the given system of equations. $\newline$ We have the following system: $\newline$ $y = -3x$ $\newline$ $\frac{y - 7}{6} = \frac{3x^2 + x - 5}{2}$
Substitute $y$ in Second Equation: Substitute the expression for $y$ from the first equation into the second equation. $\newline$ Since $y = -3x$ , we can replace $y$ in the second equation with $-3x$ : $\newline$ $\frac{(-3x - 7)}{6} = \frac{(3x^2 + x - 5)}{2}$
Eliminate Fraction: Multiply both sides of the equation by $6$ to eliminate the fraction on the left side. $\newline$ $6 \times \left(\frac{-3x - 7}{6}\right) = 6 \times \left(\frac{3x^2 + x - 5}{2}\right)$ $\newline$ This simplifies to: $\newline$ $-3x - 7 = 3 \times (3x^2 + x - 5)$
Multiply Out: Multiply out the right side of the equation. $-3x - 7 = 9x^2 + 3x - 15$
Rearrange to Quadratic Equation: Rearrange the equation to set it to zero and form a quadratic equation. $\newline$ $9x^2 + 3x - 15 + 3x + 7 = 0$ $\newline$ $9x^2 + 6x - 8 = 0$
Factor or Use Quadratic Formula: Factor the quadratic equation, if possible, or use the quadratic formula to find the roots. $\newline$ The quadratic equation $9x^2 + 6x - 8$ does not factor easily, so we will use the quadratic formula: $\newline$ $x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$ $\newline$ Here, $a = 9$ , $b = 6$ , and $c = -8$ .
Calculate Discriminant: Calculate the discriminant $b^2 - 4ac$ to ensure that there are real solutions. $\newline$ Discriminant = $b^2 - 4ac$ = $6^2 - 4(9)(-8)$ = $36 + 288$ = ${324}$ $\newline$ Since the discriminant is positive, there are two distinct real solutions.
Calculate Roots: Calculate the roots using the quadratic formula. $\newline$ $x_{1,2} = \frac{{-6 \pm \sqrt{324}}}{{2 \cdot 9}}$ $\newline$ $x_{1,2} = \frac{{-6 \pm 18}}{{18}}$ $\newline$ $x_{1} = \frac{{12}}{{18}} = \frac{2}{3}$ $\newline$ $x_{2} = \frac{{-24}}{{18}} = -\frac{4}{3}$
Find Corresponding y-values: Find the corresponding y-values using the first equation $y = -3x$ . $y_{(1)} = -3 \times \left(\frac{2}{3}\right) = -2$ $y_{(2)} = -3 \times \left(-\frac{4}{3}\right) = 4$
Calculate Sum of y-values: Calculate the sum of $y_{1} + y_{2}$ . $\newline$ $y_{1} + y_{2} = -2 + 4 = 2$