Determine the number of solutions for the given system.

One solution
No solutions
Infinitely many solutions

Ask by Turner Barker. in the United States

Mar 31,2025

Question

Determine the number of solutions for the given system. $$ y=6 x-\frac{5}{6} $$ $$ y=-6 x+5 $$ One solution No solutions Infinitely many solutions

UpStudy AI · Accepted Answer

Solve the system of equations $$ y=6x-\frac{5}{6},y=-6x+5 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}y=6x-\frac{5}{6}\y=-6x+5\end{array}ight.$$
- step1: Substitute the value of $$y:$$
  $$6x-\frac{5}{6}=-6x+5$$
- step2: Multiply both sides of the equation by LCM:
  $$\left(6x-\frac{5}{6}ight)	imes 6=\left(-6x+5ight)	imes 6$$
- step3: Calculate:
  $$36x-5=\left(-6x+5ight)	imes 6$$
- step4: Calculate:
  $$36x-5=-36x+30$$
- step5: Move the expression to the left side:
  $$36x+36x=30+5$$
- step6: Add and subtract:
  $$72x=30+5$$
- step7: Add and subtract:
  $$72x=35$$
- step8: Divide both sides:
  $$\frac{72x}{72}=\frac{35}{72}$$
- step9: Divide the numbers:
  $$x=\frac{35}{72}$$
- step10: Substitute the value of $$x:$$
  $$y=6	imes \frac{35}{72}-\frac{5}{6}$$
- step11: Calculate:
  $$y=\frac{25}{12}$$
- step12: Calculate:
  $$\left\{ \begin{array}{l}x=\frac{35}{72}\y=\frac{25}{12}\end{array}ight.$$
- step13: Check the solution:
  $$\left\{ \begin{array}{l}x=\frac{35}{72}\y=\frac{25}{12}\end{array}ight.$$
- step14: Rewrite:
  $$\left(x,yight) = \left(\frac{35}{72},\frac{25}{12}ight)$$
The system of equations has one solution, which is $$ x = \frac{35}{72} $$ and $$ y = \frac{25}{12} $$.
The system has one solution: $$ x = \frac{35}{72} $$ and $$ y = \frac{25}{12} $$.

Determine the number of solutions for the given system.

One solution
No solutions
Infinitely many solutions

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Determine the number of solutions for the given system. One solution No solutions Infinitely many solutions