Systems of Equations

1. Can you find one solution for both equations? \begin{array}{l} x + y = 12 \\ x - y = 2 \end{array}

   Hint

   Using graphing, substitution or elimination, find a point, $(x, y)$ that makes both equations true.

   Solution

   $ (7, 5) $

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2. Can you solve this system of equations using elimination? \begin{array}{l} x + 2y = 5 \\ 3x + 2y = 17 \end{array}

   Hint

   Use subtraction to eliminate the $y$ variable, then solve for $x$.

   Solution

   $ (6, -\frac{1}{2}) $

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3. Can you solve this system of equations using elimination? \begin{array}{l} 5x + 2y = -1 \\ 3x + 7y = 11 \end{array}

   Hint

   Eliminate $x$ variables by multiplying the first equation by $3$, and the second by $5$. Or, eliminate the $y$ variable by multiplying the first equation by $7$, and the second by $2$. Then subtract.

   Solution

   $ (-1, 2) $

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4. Is this an "inconsistent system"? \begin{array}{l} y = x - 1 \\ -x + y = 2 \end{array}

   Hint

   Convert both equations to the Slope-Intercept form, and compare their slopes and y-intercepts.

   Solution

   Yes, this is an “inconsistent system”. Because these lines never meet, there is no solution to this system.

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5. What is a "consistent" system of equations?

   Hint

   How many solutions must the system have?

   Solution

   It has at least one solution. In other words, graphs of the lines meet at one or more points.

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6. What is an "inconsistent" system of equations?

   Hint

   How many solutions must the system have?

   Solution

   It has no solutions. In other words, graphs of the lines never meet.

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7. What do we call a system of equations that has infinitely many solutions?

   Solution

   The system is “dependent”, meaning the graph of the lines are “coincident” (or identical).

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8. What are the three classifications of linear systems?

   Hint

   The classifications are based on three possible types of solutions.

   Solution

   1. "Consistent and Independent" (One solution)
   2. "Consistent and Dependent" (Infinite solutions)
   3. "Inconsistent" (No solutions)

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