Study: Systems of Equations

Name _____________________________________________________ Date __________________ Score _________

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.
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 $x$.
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.
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.
5. What is a “consistent” system of equations? Hint: How many solutions must the system have?
6. What is an “inconsistent” system of equations? Hint: How many solutions must the system have?
7. What do we call a system of equations that has infinitely many solutions?
8. What are the three classifications of linear systems? Hint: The classifications are based on three possible types of solutions.
Possible Points: 0

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.
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 $x$.
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.
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.
5. What is a “consistent” system of equations? Hint: How many solutions must the system have?
6. What is an “inconsistent” system of equations? Hint: How many solutions must the system have?
7. What do we call a system of equations that has infinitely many solutions?
8. What are the three classifications of linear systems? Hint: The classifications are based on the number of possible solutions (no solutions, one solution, and infinite solutions).
Possible Points: 0

$ (7, 5) $
$ (6, -\frac{1}{2}) $
$ (-1, 2) $
Yes, this is an “inconsistent system”. Because these lines never meet, there is no solution to this system.
It has at least one solution. In other words, graphs of the lines meet at one or more points.
It has no solutions. In other words, graphs of the lines never meet.
The system is “dependent”, meaning the graph of the lines are “coincident” (or identical).
1. "Consistent and Independent" (One solution)
2. "Consistent and Dependent" (Infinite solutions)
3. "Inconsistent" (No solutions)
  1. $ (7, 5) $
  2. $ (6, -\frac{1}{2}) $
  3. $ (-1, 2) $
  4. Yes, this is an “inconsistent system”. Because these lines never meet, there is no solution to this system.
  5. It has at least one solution. In other words, graphs of the lines meet at one or more points.
  6. It has no solutions. In other words, graphs of the lines never meet.
  7. The system is “dependent”, meaning the graph of the lines are “coincident” (or identical).
  8. 1. Consistent and Independent (One solution), 2) Consistent and Dependent (Infinite solutions), 3) Inconsistent (No solutions)