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Examine the six 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
Complete steps a. through e. in order to determine which pair of equations form
a system with one solution, no solution, or infinite solutions.

a. Determine which equations are in slope-intercept form. If not, then solve
the equation for y to identify the slope-intercept form.

b. Graph all six equations on the coordinate grid and label them 1-6.

c. Circle the two equations that form a system with infinite solutions.

d. Put a square around the two equations that form a system with no solution.

e. Underline the two equations that form a system with one solution.

1 answer

To analyze the six equations, we'll follow the steps outlined.

Step a: Identify or Rewrite in Slope-Intercept Form (y = mx + b)

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

    • Already in slope-intercept form.
    • Slope \( m = 5 \), y-intercept \( b = -4 \).

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

    • Already in slope-intercept form.
    • Slope \( m = -2 \), y-intercept \( b = 3 \).

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

    • Already in slope-intercept form.
    • Slope \( m = 3 \), y-intercept \( b = 1 \).

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

    • Divide by 2: \( y = 3x + 1 \).
    • Now in slope-intercept form.
    • Slope \( m = 3 \), y-intercept \( b = 1 \).

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

    • Rewrite: \( y = 5x + 2 \).
    • Now in slope-intercept form.
    • Slope \( m = 5 \), y-intercept \( b = 2 \).

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

    • Rewrite: \( y = -4x - 1 \).
    • Now in slope-intercept form.
    • Slope \( m = -4 \), y-intercept \( b = -1 \).

Summary of Slope-Intercept Forms

  • 1: \( y = 5x - 4 \) (Slope = 5)
  • 2: \( y = -2x + 3 \) (Slope = -2)
  • 3: \( y = 3x + 1 \) (Slope = 3)
  • 4: \( y = 3x + 1 \) (Slope = 3, same as Equation 3)
  • 5: \( y = 5x + 2 \) (Slope = 5)
  • 6: \( y = -4x - 1 \) (Slope = -4)

Step b: Graphing the Equations

You can graph these equations on a coordinate grid. When graphed, the lines should look like the following:

  1. Line with slope 5 and intercept -4 (steeply rising).
  2. Line with slope -2 and intercept 3 (falling).
  3. Line with slope 3 and intercept 1 (rising).
  4. Line with slope 3 and intercept 1 (overlaps with Equation 3).
  5. Line with slope 5 and intercept 2 (rising).
  6. Line with slope -4 and intercept -1 (falling).

Step c: Circle the Two Equations that Form a System with Infinite Solutions

  • Equation 3: \( y = 3x + 1 \)
  • Equation 4: \( 2y = 6x + 2 \) (reduced to \( y = 3x + 1 \))

Both equations are identical and hence form a system with infinite solutions. They lie on top of one another.

Step d: Square the Two Equations that Form a System with No Solution

  • Equation 1: \( y = 5x - 4 \)
  • Equation 5: \( y = 5x + 2 \)

Both lines have the same slope (5) but different y-intercepts (-4 for Equation 1 and 2 for Equation 5). Therefore, they are parallel and will never intersect, leading to no solution.

Step e: Underline the Two Equations that Form a System with One Solution

  • Equation 2: \( y = -2x + 3 \)
  • Equation 6: \( y = -4x - 1 \)

The lines represented by these equations have different slopes; hence, they will intersect at exactly one point, leading to one solution.

Final Summary:

  • Circles: Infinite Solutions - Equations 3 and 4
  • Squares: No Solution - Equations 1 and 5
  • Underlines: One Solution - Equations 2 and 6