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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

1 answer

Let's begin with the equations given and ensure they are all in the format \( y = mx + b \):

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

    • This is already in the required format.

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

    • This is also already in the required format.

Next, let’s rewrite the remaining equations in \( y = mx + b \) format.

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

    • This is already in the required format.

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

    • To solve for \( y \), divide the entire equation by 2: \[ y = 3x + 1 \]

  3. Equation 5: \( y - 5x = 2 \)

    • To put it in the format \( y = mx + b \), add \( 5x \) to both sides: \[ y = 5x + 2 \]

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

    • To put it in the \( y = mx + b \) format, isolate \( y \): \[ y = 4x - 1 \]

  5. Equation 7: \( y = 5x \)

    • This is already in the required format.

Now let's summarize all the equations:

  1. \( y = 5x - 4 \)
  2. \( y = -2x + 3 \)
  3. \( y = 3x + 1 \)
  4. \( y = 3x + 1 \) (from Equation 4)
  5. \( y = 5x + 2 \)
  6. \( y = 4x - 1 \)
  7. \( y = 5x \)

Now we can proceed to the next steps (b. through e.) to analyze the pairs of equations regarding the number of solutions (one solution, no solution, or infinite solutions).

Step b: Identify pairs of equations to compare.

Let’s select pairs of equations and analyze their slopes and intercepts:

  1. Pair 1: Equations 1 and 5 (\( y = 5x - 4 \) and \( y = 5x + 2 \)): These are parallel lines (same slope of 5) and therefore have no solution.

  2. Pair 2: Equations 1 and 2 (\( y = 5x - 4 \) and \( y = -2x + 3 \)): These lines intersect because they have different slopes (5 and -2), so they have one solution.

  3. Pair 3: Equations 3 and 4 (\( y = 3x + 1 \) and \( y = 3x + 1 \)): These two equations are identical (the same line), hence there are infinite solutions.

  4. Pair 4: Equations 6 and 7 (\( y = 4x -1 \) and \( y = 5x \)): These lines will intersect because they have different slopes (4 and 5), so they have one solution.

  5. Pair 5: Equations 2 and 6 (\( y = -2x + 3 \) and \( y = 4x - 1 \)): These lines will intersect (different slopes -2 and 4), so they also have one solution.

Step c: Summarize the findings:

  • Pair 1: No solution (parallel lines)
  • Pair 2: One solution (intersecting lines)
  • Pair 3: Infinite solutions (same line)
  • Pair 4: One solution (intersecting lines)
  • Pair 5: One solution (intersecting lines)

Steps d and e: Further exploration for specific pairs (if required) can be designed based on the findings above.

In conclusion, the equations can form systems with one solution, no solution, or infinite solutions depending on the pairs selected for analysis.