given the linear system:\n$5x + 3y = 17$\n$-8x - 3y = 9$\nwhat is the correct formula for the solution to…

Explanation:

Step1: Add the two equations

$(5x + 3y) + (-8x - 3y) = 17 + 9$ Simplify: $-3x = 26$

Step2: Solve for $x$

$x = \frac{26}{-3} = -\frac{26}{3}$

Step3: Substitute $x$ into first equation

$5\times\left(-\frac{26}{3}\right) + 3y = 17$ Simplify: $-\frac{130}{3} + 3y = 17$

Step4: Solve for $y$

$3y = 17 + \frac{130}{3} = \frac{51}{3} + \frac{130}{3} = \frac{181}{3}$ $y = \frac{181}{9}$

Step5: Write solution formula

Using elimination method, the solution is found by adding equations first, then substituting. The general solution steps translate to:

1. $x = \frac{(17)(-3) - (9)(3)}{(5)(-3) - (-8)(3)}$ (Cramer's Rule alternative, or direct elimination)
2. $y = \frac{(5)(9) - (-8)(17)}{(5)(-3) - (-8)(3)}$ Or via elimination: After adding equations: $-3x=26 \implies x=-\frac{26}{3}$, then substitute to find $y$.

Answer:

The solution is found by first adding the two equations to eliminate $y$: $$-3x = 26 \implies x = -\frac{26}{3}$$ Then substitute $x$ into $5x + 3y = 17$ to solve for $y$: $$y = \frac{17 - 5x}{3} = \frac{17 - 5\times\left(-\frac{26}{3}\right)}{3} = \frac{181}{9}$$ The final solution is $x = -\frac{26}{3}$, $y = \frac{181}{9}$