https://file.aiquickdraw.com/mathbot/file/1727209263104o2bev6pl.jpg

Let's break down and solve the problem shown in the image.

### Given Simultaneous Equations:

1. \( x - 9y = 10 \)
2. \( 3y^2 = 4x + 7 \)

#### Part (a) - Show that \( 3y^2 - 36y - 47 = 0 \)

We need to manipulate the two given equations to reach this form.

Start with the first equation:
\[
x = 9y + 10
\]
Now, substitute this expression for \(x\) into the second equation:
\[
3y^2 = 4x + 7
\]
Substitute \(x = 9y + 10\) into the equation:
\[
3y^2 = 4(9y + 10) + 7
\]
Expand the right-hand side:
\[
3y^2 = 36y + 40 + 7
\]
Simplify:
\[
3y^2 = 36y + 47
\]
Now, subtract \(36y + 47\) from both sides to get:
\[
3y^2 - 36y - 47 = 0
\]
Thus, part (a) is verified.

#### Part (b) - Solve the Simultaneous Equations

Now, we solve the quadratic equation \(3y^2 - 36y - 47 = 0\).

We'll use the **quadratic formula**:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \(3y^2 - 36y - 47 = 0\), the coefficients are:
- \(a = 3\)
- \(b = -36\)
- \(c = -47\)

Now, calculating the discriminant:
\[
\Delta = (-36)^2 - 4(3)(-47)
\]
\[
\Delta = 1296 + 564 = 1860
\]
Thus:
\[
y = \frac{36 \pm \sqrt{1860}}{6}
\]
Approximating \(\sqrt{1860}\):
\[
\sqrt{1860} \approx 43.13
\]
So the two possible values of \(y\) are:
\[
y_1 = \frac{36 + 43.13}{6} \approx \frac{79.13}{6} \approx 13.19
\]
\[
y_2 = \frac{36 - 43.13}{6} \approx \frac{-7.13}{6} \approx -1.19
\]

Now, substitute these values of \(y\) back into the equation \(x = 9y + 10\) to find the corresponding values of \(x\).

For \(y_1 = 13.19\):
\[
x_1 = 9(13.19) + 10 \approx 118.71 + 10 = 128.71
\]

For \(y_2 = -1.19\):
\[
x_2 = 9(-1.19) + 10 \approx -10.71 + 10 = -0.71
\]

Thus, the solutions are approximately:
\[
(x_1, y_1) \approx (128.7, 13.2)
\]
\[
(x_2, y_2) \approx (-0.7, -1.2)
\]

Would you like any further details or clarifications on these steps?

### Related Questions:
1. What are the key steps for solving quadratic equations using the quadratic formula?
2. How would the method change if the simultaneous equations involved more variables?
3. What happens if the discriminant in the quadratic formula is negative?
4. Can we solve these equations using substitution and elimination without the quadratic formula?
5. How can this method be applied to real-world problems involving simultaneous equations?

### Tip:
When solving simultaneous equations, always simplify one equation to express one variable in terms of the other before substituting into the second equation. This reduces complexity.

Simultaneous equations: x - 9y = 10 and 3y^2 = 4x + 7. Part (a): Show that 3y^2 - 36y - 47 = 0. Part (b): Solve the simultaneous equations.

Learn how to solve simultaneous equations involving a linear and quadratic equation. This problem involves the substitution method, quadratic equations, and the quadratic formula. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation