To solve the given simultaneous equations:

1. $x - 3y = 13$
2. $2x^2 - 9y = 116$

We will solve this step by step.

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Step 1: Solve Equation (1) for $x$ in terms of $y$:

From $x - 3y = 13$, isolate $x$: $x = 13 + 3y$

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Step 2: Substitute $x = 13 + 3y$ into Equation (2):

The second equation is: $2x^2 - 9y = 116$ Substitute $x = 13 + 3y$ into $x^2$: $x^2 = (13 + 3y)^2$ First, expand $(13 + 3y)^2$ using the square formula $(a + b)^2 = a^2 + 2ab + b^2$: $(13 + 3y)^2 = 13^2 + 2(13)(3y) + (3y)^2$ $(13 + 3y)^2 = 169 + 78y + 9y^2$

Now substitute $x^2 = 169 + 78y + 9y^2$ into Equation (2): $2x^2 - 9y = 116$ $2(169 + 78y + 9y^2) - 9y = 116$

Distribute the $2$: $2(169) + 2(78y) + 2(9y^2) - 9y = 116$ $338 + 156y + 18y^2 - 9y = 116$

Combine like terms: $18y^2 + 147y + 338 = 116$

Simplify by subtracting $116$ from both sides: $18y^2 + 147y + 222 = 0$

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Step 3: Solve the quadratic equation $18y^2 + 147y + 222 = 0$:

To solve this quadratic equation, we will use the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here:

• $a = 18$,
• $b = 147$,
• $c = 222$.

Step 3.1: Calculate the discriminant ($\Delta$):

$\Delta = b^2 - 4ac$ $\Delta = 147^2 - 4(18)(222)$ $\Delta = 21609 - 15984$ $\Delta = 5625$

Step 3.2: Find $\sqrt{\Delta}$:

$\sqrt{5625} = 75$

Step 3.3: Apply the quadratic formula:

$y = \frac{-b \pm \sqrt{\Delta}}{2a}$ $y = \frac{-147 \pm 75}{2(18)}$ $y = \frac{-147 \pm 75}{36}$

We split this into two cases:

1. Case 1: $y = \frac{-147 + 75}{36}$: $y = \frac{-72}{36} = -2$

2. Case 2: $y = \frac{-147 - 75}{36}$: $y = \frac{-222}{36} = -\frac{37}{6}$

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Step 4: Solve for $x$ corresponding to each $y$:

Recall that $x = 13 + 3y$.

For $y = -2$:

$x = 13 + 3(-2)$ $x = 13 - 6 = 7$

Thus, one solution is: $x = 7, \, y = -2$

For $y = -\frac{37}{6}$:

$x = 13 + 3\left(-\frac{37}{6}\right)$ Simplify: $x = 13 - \frac{111}{6}$ Convert $13$ to have a denominator of $6$: $x = \frac{78}{6} - \frac{111}{6}$ $x = \frac{-33}{6} = -\frac{11}{2}$

Thus, the second solution is: $x = -\frac{11}{2}, \, y = -\frac{37}{6}$

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Final Answer:

The solutions to the simultaneous equations are:

1. $x = 7, \, y = -2$
2. $x = -\frac{11}{2}, \, y = -\frac{37}{6}$

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Verification:

Substitute these values back into the original equations to confirm they satisfy both equations.

Let me know if you want the verification steps or have further questions!

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Related Questions:

1. Solve another pair of simultaneous equations where one equation is linear and the other is quadratic.
2. Verify the solutions for $x = 7, y = -2$ and $x = -\frac{11}{2}, y = -\frac{37}{6}$.
3. Explain the method of solving a quadratic equation using the quadratic formula.
4. Solve the simultaneous equations $x + y = 4$ and $x^2 + y^2 = 20$.
5. Derive the discriminant formula and explain its significance in quadratic equations.

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Tip:

When solving simultaneous equations with one quadratic equation, always isolate one variable in the linear equation first and substitute it into the quadratic equation. This simplifies the process.