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The graph of the solution set for the system of linear equations below is a single line in the (x,y) coordinate plane. What is the value of k? (Find what makes the two equations the same.) 12x-20y=108 3x+ky=27

User Temmy
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Final answer:

To ensure that the system of equations 12x-20y=108 and 3x+ky=27 graphs as a single line, we need to find a value of k that makes their slopes identical. By transforming both equations to slope-intercept form and equating slopes, we determine that k must equal -5.

Step-by-step explanation:

To find the value of k that will result in the system of linear equations 12x-20y=108 and 3x+ky=27 having a graph which is a single line, we need to set the two equations to have the same slope, since the slope is constant along a straight line. We can use the slope-intercept form, y=mx+b, where m is the slope and b is the y-intercept.

First, let's write the first equation in the slope-intercept form:

  1. Divide every term by 4: 3x-5y=27
  2. Isolate y: y = \(\frac{3}{5}x - \frac{27}{5}\)

The slope of this line is 3/5 and the y-intercept is not relevant for this problem as we are only concerned with the slope for equating two lines.

Now we'll rewrite the second equation to find its slope:

  1. Isolate y: y = \(-\frac{3}{k}x + \frac{27}{k}\)

For the two lines to coincide, their slopes must be equal, which means:

\(\frac{3}{5} = -\frac{3}{k}\)

By solving the above equation, we get k = -5. Thus, when k is -5, both equations represent the same line in the coordinate plane, meeting the condition provided.

User Feigi
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