Solve the simultaneous equations:

y = 9 - 2x

y2 + x2 = 17

If necessary, give answers as fractions in simplest for. For example: -3/2

For y > 2 the solution is: 

x =

Answer 1 Question 5

and y=

Answer 2 Question 5

The other solution is: 

x =

Answer 3 Question 5

and y=

Answer 4 Question 5


We are tasked with solving the following system of simultaneous equations:

1. \( y = 9 - 2x \)
2. \( y^2 + x^2 = 17 \)

### Step 1: Substituting for \( y \)

Since we know that \( y = 9 - 2x \), we can substitute this expression into the second equation:

\[
(9 - 2x)^2 + x^2 = 17
\]

### Step 2: Expanding the equation

Expand \( (9 - 2x)^2 \):

\[
(9 - 2x)^2 = 81 - 36x + 4x^2
\]

Now, substitute this back into the equation:

\[
81 - 36x + 4x^2 + x^2 = 17
\]

### Step 3: Simplify the equation

Combine like terms:

\[
5x^2 - 36x + 81 = 17
\]

Subtract 17 from both sides:

\[
5x^2 - 36x + 64 = 0
\]

### Step 4: Solving the quadratic equation

We now have the quadratic equation:

\[
5x^2 - 36x + 64 = 0
\]

We'll solve this using the quadratic formula, which is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 5 \), \( b = -36 \), and \( c = 64 \). Substituting these values into the formula:

\[
x = \frac{-(-36) \pm \sqrt{(-36)^2 - 4(5)(64)}}{2(5)}
\]

\[
x = \frac{36 \pm \sqrt{1296 - 1280}}{10}
\]

\[
x = \frac{36 \pm \sqrt{16}}{10}
\]

\[
x = \frac{36 \pm 4}{10}
\]

### Step 5: Find the two solutions for \( x \)

Now, we calculate the two possible values for \( x \):

1. \( x = \frac{36 + 4}{10} = \frac{40}{10} = 4 \)
2. \( x = \frac{36 - 4}{10} = \frac{32}{10} = \frac{16}{5} \)

### Step 6: Find the corresponding values of \( y \)

Using the equation \( y = 9 - 2x \), we find the corresponding values of \( y \):

1. If \( x = 4 \):

\[
y = 9 - 2(4) = 9 - 8 = 1
\]

2. If \( x = \frac{16}{5} \):

\[
y = 9 - 2\left(\frac{16}{5}\right) = 9 - \frac{32}{5} = \frac{45}{5} - \frac{32}{5} = \frac{13}{5}
\]

### Step 7: Conclusion

For \( y > 2 \), the solution is:

- \( x = \frac{16}{5} \)
- \( y = \frac{13}{5} \)

The other solution is:

- \( x = 4 \)
- \( y = 1 \)

### Do you want more details or have any questions?

Here are five related questions:

1. How do you derive the quadratic formula?
2. What are other methods to solve simultaneous equations?
3. How can you tell which solution satisfies \( y > 2 \)?
4. What are the applications of simultaneous equations in real life?
5. How can you check if a solution is valid for both equations?

**Tip**: Always verify your solutions by substituting them back into the original equations.

Solve the simultaneous equations: y = 9 - 2x and y^2 + x^2 = 17. If necessary, give answers as fractions in simplest form. For example: -3/2. For y > 2 the solution is: x = Answer 1 Question 5 and y = Answer 2 Question 5. The other solution is: x = Answer 3 Question 5 and y = Answer 4 Question 5.

Learn how to solve the simultaneous equations y = 9 - 2x and y² + x² = 17 using substitution and the quadratic formula. This step-by-step solution covers key algebraic methods and is suitable for students in Grades 8-10.

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