Question

$4x^{2}+9y^{2}=72$
$2x - y = 4$
which of the following is a solution to the system shown?
(2, 3)
(14/5, -3/5)
(-3, -2)
(3/5, -14/5)

Explanation:

Step1: Isolate y from linear equation

From $2x - y = 4$, rearrange to get $y = 2x - 4$.

Step2: Substitute y into ellipse equation

Substitute $y = 2x - 4$ into $4x^2 + 9y^2 = 72$:
$$4x^2 + 9(2x - 4)^2 = 72$$
Expand $(2x-4)^2$:
$$4x^2 + 9(4x^2 - 16x + 16) = 72$$
$$4x^2 + 36x^2 - 144x + 144 = 72$$
Combine like terms:
$$40x^2 - 144x + 72 = 0$$
Divide all terms by 8:
$$5x^2 - 18x + 9 = 0$$

Step3: Solve quadratic equation

Use quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=5$, $b=-18$, $c=9$:
$$x = \frac{18 \pm \sqrt{(-18)^2 - 4(5)(9)}}{2(5)}$$
$$x = \frac{18 \pm \sqrt{324 - 180}}{10}$$
$$x = \frac{18 \pm \sqrt{144}}{10}$$
$$x = \frac{18 \pm 12}{10}$$
Calculate two solutions:
$x = \frac{18+12}{10} = 3$ or $x = \frac{18-12}{10} = \frac{6}{10} = \frac{3}{5}$

Step4: Find corresponding y values

For $x=3$: $y = 2(3) - 4 = 2$
For $x=\frac{3}{5}$: $y = 2(\frac{3}{5}) - 4 = \frac{6}{5} - \frac{20}{5} = -\frac{14}{5}$

Step5: Match to options

The solution $(\frac{3}{5}, -\frac{14}{5})$ is one of the choices.

Answer:

D. (3/5, -14/5)