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Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. {:[-x+2y=-4],[-2x+7y=-6]:} Infinitely Many Solutions One Solution No Solutions

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline−x+2y=−4−2x+7y=−6 \begin{array}{r} -x+2 y=-4 \\ -2 x+7 y=-6 \end{array} \newlineInfinitely Many Solutions\newlineOne Solution\newlineNo Solutions

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Q. Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.\newline−x+2y=−4−2x+7y=−6 \begin{array}{r} -x+2 y=-4 \\ -2 x+7 y=-6 \end{array} \newlineInfinitely Many Solutions\newlineOne Solution\newlineNo Solutions
  1. Write Equations: Write down the system of equations.\newlineThe system of equations is given as:\newline−1x+2y=−4-1x + 2y = -4\newline−2x+7y=−6-2x + 7y = -6\newlineWe need to determine the number of solutions this system has.
  2. Elimination Method: Attempt to solve the system using the elimination method.\newlineFirst, we can multiply the first equation by 22 to make the coefficients of xx in both equations the same.\newline2(−1x+2y)=2(−4)2(-1x + 2y) = 2(-4)\newlineThis gives us:\newline−2x+4y=−8-2x + 4y = -8\newlineNow we have the system:\newline−2x+4y=−8-2x + 4y = -8\newline−2x+7y=−6-2x + 7y = -6
  3. Subtract Equations: Subtract the first new equation from the second equation to eliminate xx. (−2x+7y)−(−2x+4y)=−6−(−8)\ (-2x + 7y) - (-2x + 4y) = -6 - (-8)This simplifies to: 7y−4y=−6+8\ 7y - 4y = -6 + 8 3y=2\ 3y = 2
  4. Solve for y: Solve for y.\newlineDivide both sides of the equation by 33 to isolate yy.\newline3y3=23\frac{3y}{3} = \frac{2}{3}\newliney=23y = \frac{2}{3}
  5. Substitute and Solve: Substitute the value of yy back into one of the original equations to solve for xx. Using the first original equation: −1x+2(23)=−4-1x + 2(\frac{2}{3}) = -4 −1x+43=−4-1x + \frac{4}{3} = -4 Multiply both sides by 33 to clear the fraction: −3x+4=−12-3x + 4 = -12
  6. Solve for x: Solve for x.\newlineSubtract 44 from both sides:\newline−3x=−12−4-3x = -12 - 4\newline−3x=−16-3x = -16\newlineDivide both sides by −3-3:\newlinex=−16−3x = \frac{-16}{-3}\newlinex=163x = \frac{16}{3}
  7. Check Solution: Check the solution by substituting xx and yy back into the second original equation.\newline−2x+7y=−6-2x + 7y = -6\newline−2(163)+7(23)=−6-2(\frac{16}{3}) + 7(\frac{2}{3}) = -6\newline−323+143=−6-\frac{32}{3} + \frac{14}{3} = -6\newline−183=−6-\frac{18}{3} = -6\newline−6=−6-6 = -6\newlineThe solution checks out.

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