How many solutions does the linear system y = 2x + 5 and 8x – 4y = 20 have?

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To determine how many solutions the linear system has, we need to analyze the two equations:

  • Equation 1: y = 2x + 5
  • Equation 2: 8x – 4y = 20

First, let’s rearrange Equation 2 to express it in slope-intercept form (y = mx + b) similar to Equation 1. We want to isolate y:

8x - 4y = 20 
=> -4y = -8x + 20 
=> y = 2x - 5

Now we have:

  • Equation 1: y = 2x + 5
  • Equation 2: y = 2x – 5

Next, we can observe that both equations are straight lines with the same slope (m = 2). In geometric terms, lines that have the same slope are either:

  • Parallel lines: If the lines are parallel, they will never intersect and hence have no solutions.
  • Identical lines: If the lines are the same, they will have infinitely many solutions.

To check if they are identical or just parallel, we can compare their y-intercepts. From Equation 1, the y-intercept is +5, while from Equation 2, the y-intercept is -5. Since these lines have different y-intercepts, they cannot be the same line.

Thus, since both equations are parallel with the same slope but different y-intercepts, the conclusion is:

Final Answer

The linear system has no solutions.

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