8.2: How Many Solutions?
This activity is intended to supplement Algebra I, Chapter 7, Lesson 5.
In this activity, you will explore:
- Graphing linear systems to determine the number of solutions
- Creating a linear system with a particular number of solutions
- The relationship between the coefficients of a linear system and the number of solutions
Problem 1 - Graphing systems of linear equations
Do all linear systems have just one solution? In this problem, you will graph several different linear systems to see how many solutions they have.
Graph each system below by solving the equations for \begin{align*}y\end{align*} and entering them into \begin{align*}Y1\end{align*} and \begin{align*}Y2\end{align*}. View the graph in a standard viewing window. (If necessary, set the viewing window by going to Zoom > ZStandard.)
Sketch each graph. How many solutions does each system have?
1. \begin{align*}\begin{cases} y = 2x - 3 \\ y = x - 1 \\ \end{cases}\end{align*}
2. \begin{align*}\begin{cases} y = -3x + 3 \\ y = -3x - 1 \\ \end{cases}\end{align*}
3. \begin{align*}\begin{cases} 4x + 2y = 6 \\ y = -2x + 3 \\ \end{cases}\end{align*}
Sketch:
Number of solutions:
Sketch:
Number of solutions:
Sketch:
Number of solutions:
These three graphs show all the possible ways two lines can relate to each other.
\begin{align*} & \text{If the two lines} \ldots \qquad \qquad \qquad \quad \ \text{Then the system has} \ldots \\ & \bullet \text{Cross at a single point} \qquad \longrightarrow \quad \ \bullet \ \text{One solution} \\ & \bullet \text{Never cross (are parallel)} \quad \longrightarrow \ \quad \bullet \ \text{No solution} \\ & \bullet \text{Are really the same line} \ \ \quad \longrightarrow \ \quad \bullet \ \text{Infinitely many solutions}\end{align*}
Problem 2 - Create your own system
Can you create a system with one solution?
The CabriJr file HOWMANY1 shows the graph of a linear equation. Press [ALPHA] \begin{align*}F1\end{align*} and scroll down to Open . . .
Use the Line tool ([ALPHA] \begin{align*}F2\end{align*} ENTER Line ENTER) to draw a second line on the graph so that the two lines form a linear system with exactly one solution.
Use the Coordinates and Equations tool ([ALPHA] \begin{align*}F5\end{align*} ENTER Coord. & Eq. ENTER) to find the equations of the two lines. Record them in the table.
Delete the line you drew. Draw a new line to make a system with no solution. Record the equation of the line in the table.
Next, delete that line. Draw a new line to make a system with infinitely many solutions. Record the equation of the line.
Repeat this experiment with the lines you find in the CabriJr files HOWMANY2, HOWMANY3, and HOWMANY4. For each file, make a system with one solution, a system with no solutions, and a system with infinitely many solutions. Record all the equations in the table.
Original Line | One Solution | No Solutions | Infinitely Many Solutions |
---|---|---|---|
HOWMANY1
y = |
\begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} | \begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} | \begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} |
HOWMANY2
y = |
\begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} | \begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} | \begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} |
HOWMANY3
y = |
\begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} | \begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} | \begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} |
HOWMANY4
y = |
\begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} | \begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} | \begin{align*}\begin{cases} y = \\ y = \\ \end{cases}\end{align*} |
Compare the equations for the lines you drew with the equations of the original line that was drawn for you.
4. Which equations have the same slope as the original equation? Those that form a system with one solution, no solution, or many solutions?
5. Which equations have the same \begin{align*}y-\end{align*}intercept as the original equation? Those that form a system with one solution, no solution, or many solutions?
6. Which equations are equivalent to the original equation?
7. Why is it sometimes hard to see that two equations in a linear system are equivalent? Give an example.
8. Complete each statement to create some rules about the number of solutions for a linear system of equations.
- A linear system has no solution if the equations have ____ slopes and ____ \begin{align*}y-\end{align*}intercepts.
- A linear system has infinitely many solutions if the equations have ____ slopes and ____ \begin{align*}y-\end{align*}intercepts.
- A linear system has one solution if the equations have ____ slopes and ____ \begin{align*}y-\end{align*}intercepts.
Determine how many solutions each system has without graphing.
9. \begin{align*}\begin{cases} y = x \\ y = 2x\\ \end{cases}\end{align*}
10. \begin{align*}\begin{cases} 3x + 4y = 12 \\ 2x + 4y = 8 \\ \end{cases}\end{align*}
11. \begin{align*}\begin{cases} y = \frac{1}{2}x + 1 \\ y = \frac{1}{2}x + 8 \\ \end{cases}\end{align*}
12. \begin{align*}\begin{cases} y = \frac{1}{2}x + 2 \\ -2y = - x - 4 \\ \end{cases}\end{align*}
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