Question
upstudy study bank question image url

4 SIMULTANEOUS 틀 Activity 5 Solve for \( x \) and \( y \) in: \( 5.1 \quad 2 y=3+x \) and \( 7=x^{2}+4 y^{2} \)

Ask by Hilton Ramirez. in South Africa
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The solutions are: \[ \left( x, y \right) = \left( \frac{-3+\sqrt{5}}{2}, \frac{3+\sqrt{5}}{4} \right) \quad \text{and} \quad \left( x, y \right) = \left( \frac{-3-\sqrt{5}}{2}, \frac{3-\sqrt{5}}{4} \right) \]

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Beyond the Answer

To solve the simultaneous equations \( 2y = 3 + x \) and \( 7 = x^2 + 4y^2 \), we can start by expressing \( y \) in terms of \( x \) from the first equation. From \( 2y = 3 + x \), we can express \( y \) as: \[ y = \frac{3 + x}{2} \] Next, we substitute this expression for \( y \) into the second equation: \[ 7 = x^2 + 4\left(\frac{3 + x}{2}\right)^2 \] Now, simplifying the second equation: \[ 7 = x^2 + 4\left(\frac{(3 + x)^2}{4}\right) \] \[ 7 = x^2 + (3 + x)^2 \] \[ = x^2 + (9 + 6x + x^2) \] \[ = 2x^2 + 6x + 9 \] Setting this equal to 7 gives: \[ 7 = 2x^2 + 6x + 9 \] \[ 0 = 2x^2 + 6x + 9 - 7 \] \[ 0 = 2x^2 + 6x + 2 \] Dividing the whole equation by 2, we get: \[ 0 = x^2 + 3x + 1 \] Now we can use the quadratic formula to solve for \( x \): \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} = \frac{{-3 \pm \sqrt{{3^2 - 4 \cdot 1 \cdot 1}}}}{2 \cdot 1} \] \[ = \frac{{-3 \pm \sqrt{9 - 4}}}{2} \] \[ = \frac{{-3 \pm \sqrt{5}}}{2} \] So, the solutions for \( x \) are: \[ x_1 = \frac{{-3 + \sqrt{5}}}{2}, \quad x_2 = \frac{{-3 - \sqrt{5}}}{2} \] Next, we substitute these values back into the equation for \( y \): For \( x_1 \): \[ y_1 = \frac{3 + \frac{{-3 + \sqrt{5}}}{2}}{2} = \frac{3 - 3 + \sqrt{5}}{4} = \frac{\sqrt{5}}{4} \] For \( x_2 \): \[ y_2 = \frac{3 + \frac{{-3 - \sqrt{5}}}{2}}{2} = \frac{3 - 3 - \sqrt{5}}{4} = \frac{-\sqrt{5}}{4} \] Thus, the solutions are: \[ (x_1, y_1) = \left(\frac{-3 + \sqrt{5}}{2}, \frac{\sqrt{5}}{4}\right), \quad (x_2, y_2) = \left(\frac{-3 - \sqrt{5}}{2}, \frac{-\sqrt{5}}{4}\right) \]

Related Questions

Latest Algebra Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy