Question
upstudy study bank question image url

Solve the system of equations. Express numbers as integers or simplified fractions. \[ \begin{aligned} 6 x+9 y & =-4 \\ 12 x+5 z & =0 \\ 6 y+5 z & =-8\end{aligned} \]

Ask by Lyons Gough. in the United States
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The solution to the system of equations is \( x = \frac{1}{3} \), \( y = -\frac{2}{3} \), and \( z = -\frac{4}{5} \).

Solution

Solve the system of equations \( 6x+9y=-4;12x+5z=0;6y+5z=-8 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}6x+9y=-4\\12x+5z=0\\6y+5z=-8\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=-\frac{4+9y}{6}\\12x+5z=0\\6y+5z=-8\end{array}\right.\) - step2: Substitute the value of \(x:\) \(\left\{ \begin{array}{l}12\left(-\frac{4+9y}{6}\right)+5z=0\\6y+5z=-8\end{array}\right.\) - step3: Simplify: \(\left\{ \begin{array}{l}-8-18y+5z=0\\6y+5z=-8\end{array}\right.\) - step4: Solve the equation: \(\left\{ \begin{array}{l}y=\frac{-8+5z}{18}\\6y+5z=-8\end{array}\right.\) - step5: Substitute the value of \(y:\) \(6\times \frac{-8+5z}{18}+5z=-8\) - step6: Multiply the terms: \(\frac{-8+5z}{3}+5z=-8\) - step7: Multiply both sides of the equation by LCD: \(\left(\frac{-8+5z}{3}+5z\right)\times 3=-8\times 3\) - step8: Simplify the equation: \(-8+20z=-24\) - step9: Move the constant to the right side: \(20z=-24+8\) - step10: Add the numbers: \(20z=-16\) - step11: Divide both sides: \(\frac{20z}{20}=\frac{-16}{20}\) - step12: Divide the numbers: \(z=-\frac{4}{5}\) - step13: Substitute the value of \(z:\) \(y=\frac{-8+5\left(-\frac{4}{5}\right)}{18}\) - step14: Calculate: \(y=-\frac{2}{3}\) - step15: Substitute the value of \(y:\) \(x=-\frac{4+9\left(-\frac{2}{3}\right)}{6}\) - step16: Substitute back: \(x=\frac{-4+9\times \frac{2}{3}}{6}\) - step17: Calculate: \(x=\frac{1}{3}\) - step18: Calculate: \(\left\{ \begin{array}{l}x=\frac{1}{3}\\y=-\frac{2}{3}\\z=-\frac{4}{5}\end{array}\right.\) - step19: Check the solution: \(\left\{ \begin{array}{l}x=\frac{1}{3}\\y=-\frac{2}{3}\\z=-\frac{4}{5}\end{array}\right.\) - step20: Rewrite: \(\left(x,y,z\right) = \left(\frac{1}{3},-\frac{2}{3},-\frac{4}{5}\right)\) The solution to the system of equations is \( (x,y,z) = \left(\frac{1}{3},-\frac{2}{3},-\frac{4}{5}\right) \) or \( (x,y,z) = (0.\dot{3},-0.\dot{6},-0.8) \).

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Extra Insights

To solve the system of equations: 1. **First, simplify the first equation:** \[ 6x + 9y = -4. \] Divide every term by 3: \[ 2x + 3y = -\frac{4}{3}. \, (1) \] 2. **Next, simplify the second equation:** \[ 12x + 5z = 0. \] This can be rewritten as: \[ z = -\frac{12}{5}x. \, (2) \] 3. **Substituting (2) into the third equation:** \[ 6y + 5z = -8. \] Substituting \(z\): \[ 6y + 5\left(-\frac{12}{5}x\right) = -8. \] Simplifying: \[ 6y - 12x = -8. \] Dividing the entire equation by 6: \[ y - 2x = -\frac{4}{3}. \, (3) \] 4. **Now we have two simplified equations:** - From (1), \(2x + 3y = -\frac{4}{3}\). - From (3), \(y - 2x = -\frac{4}{3}\). 5. **Now let's solve for \(y\) using (3):** Rearranging (3): \[ y = 2x - \frac{4}{3}. \] 6. **Substituting \(y\) back into equation (1):** \[ 2x + 3\left(2x - \frac{4}{3}\right) = -\frac{4}{3}. \] This simplifies to: \[ 2x + 6x - 4 = -\frac{4}{3}. \] Combine like terms: \[ 8x - 4 = -\frac{4}{3}. \] Add 4 to both sides: \[ 8x = 4 - \frac{4}{3} = \frac{12}{3} - \frac{4}{3} = \frac{8}{3}. \] Dividing by 8: \[ x = \frac{8}{3 \cdot 8} = \frac{1}{3}. \] 7. **Now substitute \(x\) back to find \(y\):** Substitute \(x = \frac{1}{3}\) into \(y = 2x - \frac{4}{3}\): \[ y = 2\left(\frac{1}{3}\right) - \frac{4}{3} = \frac{2}{3} - \frac{4}{3} = -\frac{2}{3}. \] 8. **Substituting \(x\) into (2) to find \(z\):** From (2): \[ z = -\frac{12}{5}\left(\frac{1}{3}\right) = -\frac{12}{15} = -\frac{4}{5}. \] Finally, the solution to the system of equations is: \[ x = \frac{1}{3}, \quad y = -\frac{2}{3}, \quad z = -\frac{4}{5}. \]

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