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{:[7p=9(p+q)+11],[9q+3=-4(7q+p)]:} Consider the system of equations. If (p,q) is the solution to the system, then what is the value of -p-q ?

7pamp;=9(p+q)+119q+3amp;=−4(7q+p) \begin{aligned} 7 p & =9(p+q)+11 \\ 9 q+3 & =-4(7 q+p) \end{aligned} \newlineConsider the system of equations. If (p,q) (p, q) is the solution to the system, then what is the value of −p−q -p-q ?

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Full solution

Q. 7p=9(p+q)+119q+3=−4(7q+p) \begin{aligned} 7 p & =9(p+q)+11 \\ 9 q+3 & =-4(7 q+p) \end{aligned} \newlineConsider the system of equations. If (p,q) (p, q) is the solution to the system, then what is the value of −p−q -p-q ?
  1. Simplify first equation: Let's start by simplifying the first equation in the system:\newline7p=9(p+q)+117p = 9(p + q) + 11\newlineDistribute the 99 on the right side:\newline7p=9p+9q+117p = 9p + 9q + 11\newlineNow, let's move all terms involving pp to one side and the constant to the other side:\newline9q+11=9p−7p9q + 11 = 9p - 7p\newline9q+11=2p9q + 11 = 2p\newlineDivide both sides by 22 to solve for pp:\newline(9q+11)/2=p(9q + 11)/2 = p
  2. Simplify second equation: Next, let's simplify the second equation in the system:\newline9q+3=−4(7q+p)9q + 3 = -4(7q + p)\newlineDistribute the −4-4 on the right side:\newline9q+3=−28q−4p9q + 3 = -28q - 4p\newlineNow, let's move all terms involving qq to one side and the constant to the other side:\newline9q+28q=−4p−39q + 28q = -4p - 3\newline37q=−4p−337q = -4p - 3\newlineDivide both sides by 3737 to solve for qq:\newlineq=−4p−337q = \frac{-4p - 3}{37}
  3. Substitute and solve for pp: Now we have two expressions for pp and qq:
    p=9q+112p = \frac{9q + 11}{2}
    q=−4p−337q = \frac{-4p - 3}{37}
    We need to substitute one of these into the other to find the values of pp and qq. Let's substitute the expression for qq into the expression for pp:
    p=9(−4p−337)+112p = \frac{9\left(\frac{-4p - 3}{37}\right) + 11}{2}
    Now, let's simplify this equation to solve for pp:
    pp11
    pp22
    pp33
    Multiply both sides by pp44 to clear the denominator:
    pp55
    Now, let's move all terms involving pp to one side:
    pp77
    pp88
    Divide both sides by pp99 to solve for pp:
    qq11
    qq22
    qq33
  4. Substitute and solve for q: Now that we have the value of pp, let's substitute it back into the expression for qq:
    q=(−4(38/11)−3)/37q = (-4(38/11) - 3)/37
    q=(−152/11−3)/37q = (-152/11 - 3)/37
    q=(−152/11−33/11)/37q = (-152/11 - 33/11)/37
    q=(−185/11)/37q = (-185/11)/37
    Multiply the numerator and denominator by the reciprocal of the denominator to simplify:
    q=−185/(11∗37)q = -185/(11*37)
    q=−185/407q = -185/407
    q=−5/11q = -5/11
    q=−0.454545...q = -0.454545...
  5. Find value of −p−q-p-q: Finally, we need to find the value of −p−q-p-q:
    −p−q=−(3811)−(−511)-p-q = -(\frac{38}{11}) - (-\frac{5}{11})
    Combine the terms:
    −p−q=−3811+511-p-q = -\frac{38}{11} + \frac{5}{11}
    −p−q=(−38+511)-p-q = (\frac{-38 + 5}{11})
    −p−q=−3311-p-q = -\frac{33}{11}
    −p−q=−3-p-q = -3

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