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Graphing Linear Equations Portfolio

Worksheet
Directions: Use this worksheet to record your answers to the graphing linear
equations activities in this portfolio. When you are finished, save this worksheet
with your answers and submit it for a portfolio grade.
Investigate and explain the different types of graphical solutions.
Question 1
Examine the different equations that follow:
1. y=5x–4
2. y=–2x+3
3. y=3x+1
4. 2y=6x+2
5. y–5x=+2
6. y–4x+1=0
7. y=5x
Complete steps a. through e. to determine which pair of equations form a system
with one solution, no solution, or infinite solutions.
a. Solve each equation and make sure it is in y=mx+b format.
1. y=5x–4
2. y=–2x+3
3. y=3x+1
4. 2y=6x+2
5. y–5x=+2
6. y–4x+1=0
7. y

1 answer

Sure! Let's go through each equation and convert them into the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step a: Rewrite the Equations in \(y = mx + b\) Format

  1. \(y = 5x - 4\)

    • This equation is already in slope-intercept form.

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

    • This equation is also already in slope-intercept form.

  3. \(y = 3x + 1\)

    • This equation is already in slope-intercept form.

  4. \(2y = 6x + 2\)

    • To convert to \(y = mx + b\): \[ y = \frac{6}{2}x + \frac{2}{2} = 3x + 1 \]

  5. \(y - 5x = 2\)

    • Rearranging gives: \[ y = 5x + 2 \]

  6. \(y - 4x + 1 = 0\)

    • Rearranging gives: \[ y = 4x - 1 \]

  7. \(y = 5x\)

    • This equation is already in slope-intercept form.

Summary of the Equations in \(y = mx + b\) Format

  1. \(y = 5x - 4\)
  2. \(y = -2x + 3\)
  3. \(y = 3x + 1\)
  4. \(y = 3x + 1\) (same as equation 3)
  5. \(y = 5x + 2\)
  6. \(y = 4x - 1\)
  7. \(y = 5x\)

Step b: Identify Pairs of Equations for Systems

Now, we can proceed to identify which pairs of equations can form a system with one solution, no solution, or infinite solutions based on their slopes and intercepts.

  1. Equations 1 & 7:

    • \(y = 5x - 4\) and \(y = 5x\) (same slope, different y-intercepts) → No solution (parallel lines)

  2. Equations 2 & 3:

    • \(y = -2x + 3\) and \(y = 3x + 1\) (different slopes) → One solution (intersecting lines)

  3. Equations 4 & 3:

    • \(y = 3x + 1\) and \(y = 3x + 1\) (same slope and intercept) → Infinite solutions (same line)

  4. Equations 5 & 1:

    • \(y = 5x + 2\) and \(y = 5x - 4\) (same slope, different y-intercepts) → No solution (parallel lines)

  5. Equations 6 & 3:

    • \(y = 4x - 1\) and \(y = 3x + 1\) (different slopes) → One solution (intersecting lines)

Step c: Graphing

When graphing these equations, the characteristics outlined will show:

  • One Solution: Where two lines intersect.
  • No Solution: Where two lines are parallel (never intersect).
  • Infinite Solutions: Where two lines lie on top of each other (exactly the same).

Step d: Conclusion

You can summarize the nature of systems formed by the given pairs:

  • 1 & 7: No solution
  • 2 & 3: One solution
  • 4 & 3: Infinite solutions
  • 5 & 1: No solution
  • 6 & 3: One solution

Step e: Reflection

Reflect on the concepts of parallel lines, identical lines, and intersecting lines. These concepts illustrate how linear equations can graphically represent relationships between different variables.

Please Proceed

If there are more questions or additional activities, feel free to ask!