Practice Ordered Pair Solutions to Linear Equations

1. Which ordered pair is a solution of the equation?

   \( y=3x+5\)
   

   Choose 1 answer:

   1. Only (2, 11)
   2. Only (3, 13)
   3. Both (2, 11) and (3, 13)
   4. Neither

2. Which ordered pair is a solution of the equation?

   \(y+1=3(x-4)\)
   

   Choose 1 answer:

   1. Only (4, -1)
   2. Only (5 , 2)
      
   3. Both (4, -1) and (5 , 2)
      

   4. Neither

3. Which ordered pair is a solution of the equation?

   \( y=7x-2\)
   

   Choose 1 answer:

   1. Only (3, 15)
   2. Only (-1 , -10)
      
   3. Both (3, 15) and (-1 , -10)
      

   4. Neither

4. Which ordered pair is a solution of the equation?

   \( -3x+5y=2x+3y\)
   

   Choose 1 answer:

   1. Only (2, 4)
   2. Only (3 ,3)
      
   3. Both (2, 5) and (3 ,3)
      
   4. Neither

Source: Khan Academy, https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:linear-equations-graphs/x2f8bb11595b61c86:two-variable-linear-equations-intro/e/plugging_in_values
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1. To check whether an ordered pair (a, b) is a solution of an equation, substitute these values into the equation and determine if the resulting equality is true or false.

   To check whether (2, 11) is a solution of the equation, let's substitute ‍\(x=2\) and \(y=11\) into the equation:

   \(\begin{aligned}{y}&=3{x}+5\\
   {11}&=3\cdot{2}+5\\
   11&=6+5\\
   11&=11\end{aligned}\)
   

   Since \(11=11\), we obtained a true statement, so (2,11) is indeed a solution of the equation.

   To check whether (3, 13) is a solution of the equation, let's substitute ‍\(x=3\) and \(y=13\) into the equation:

   \(\begin{aligned}{y}&=3{x}+5\\
   {13}&=3\cdot{3}+5\\
   13&=9+5\\
   13&=14\end{aligned}\)
   

   Since \(13 \neq 14\), we obtained a false statement, so (3, 13) is not a solution of the equation.

   Only (2,11) is a solution of the equation.

2. To check whether an ordered pair (a, b) is a solution of an equation, substitute these values into the equation and determine if the resulting equality is true or false.

   To check whether (4, -1) is a solution of the equation, let's substitute ‍\(x=4\) and \(y=-1\) into the equation:

   \(\begin{aligned}{y}+1&=3({x}-4)\\
   {-1}+1&=3({4}-4)\\
   0&=3\cdot 0\\
   0&=0\end{aligned}\)
   

   Since \(0=0\), we obtained a true statement, so (4, -1) is indeed a solution of the equation.

   To check whether (5, 2) is a solution of the equation, let's substitute ‍\(x=5\) and \(y=2\) into the equation:

   \(\begin{aligned}{y}+1&=3({x}-4)\\
   {2}+1&=3({5}-4)\\
   3&=3\cdot 1\\
   3&=3\end{aligned}\)
   

   Since \(3=3\), we obtained a true statement, so (5, 2) is indeed a solution of the equation.

   Both ordered pairs are solutions of the equation.

3. To check whether an ordered pair (a, b) is a solution of an equation, substitute these values into the equation and determine if the resulting equality is true or false.

   To check whether (3, 15) is a solution of the equation, let's substitute ‍\(x=3\) and \(y=15\) into the equation:

   \(\begin{aligned}{y}&=7{x}-2\\
   {15}&=7\cdot {3}-2\\
   15&=21-2\\
   15&=19\end{aligned}\)
   

   Since \(15 \neq 19\), we obtained a false statement, so (3, 15) is not a solution of the equation.

   To check whether (-1, 10) is a solution of the equation, let's substitute ‍\(x=-1\) and \(y=-10\) into the equation:

   \(\begin{aligned}{y}&=7{x}-2\\
   {-10}&=7\cdot({-1})-2\\
   -10&=-7-2\\
   -10&=-9\end{aligned}\)
   

   Since \(-10 \neq -9\), we obtained a false statement, so (-1, -10) is not a solution of the equation.

   Neither of the ordered pairs is a solution of the equation.

4. To check whether an ordered pair (a, b) is a solution of an equation, substitute these values into the equation and determine if the resulting equality is true or false.

   To check whether (2, 4) is a solution of the equation, let's substitute ‍\(x=2\) and \(y=4\) into the equation:

   \(\begin{aligned}-3{x}+5{y}&=2{x}+3{y}\\
   -3\cdot{2}+5\cdot{4}&=2\cdot{2}+3\cdot{4}\\
   -6+20&=4+12\\
   14&=16\end{aligned}\)
   

   Since \(14 \neq 16\), we obtained a false statement, so (2, 4) is not a solution of the equation.

   To check whether (3, 3) is a solution of the equation, let's substitute ‍\(x=3\) and \(y=3\) into the equation:

   \(\begin{aligned}-3{x}+5{y}&=2{x}+3{y}\\
   -3\cdot{3}+5\cdot{3}&=2\cdot{3}+3\cdot{3}\\
   -9+15&=6+9\\
   6&=15\end{aligned}\)
   

   Since \(6 \neq 15\), we obtained a false statement, so (3, 3) is not a solution of the equation.

   Neither of the ordered pairs is a solution of the equation.