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  1. One equation of a linear system is «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mi»y«/mi»«mo»-«/mo»«mn»1«/mn»«mo»=«/mo»«mn»0«/mn»«/math». Write a second equation that will produce a system with

    1. 1 solution

      Equations will vary. A sample is shown.

      The given line, along with any line with a different slope, will produce a system with one solution. Rearrange the given equation to slope-intercept form to determine its slope.



      The line «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mn mathcolor=¨#B94A48¨»5«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«/math» has a different slope than the line «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»6«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo mathcolor=¨#B94A48¨»=«/mo»«mn mathcolor=¨#B94A48¨»0«/mn»«/math», so the two form a system of linear equations with one solution.

    2. 0 solutions

      Equations will vary. A sample is shown.

      The line «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mfrac mathcolor=¨#B94A48¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»5«/mn»«/math» has the same slope as «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»6«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo mathcolor=¨#B94A48¨»=«/mo»«mn mathcolor=¨#B94A48¨»0«/mn»«/math», but a different
       y-intercept. The lines are parallel, so the system will have no solutions.

    3. an infinite number of solutions

      Equations will vary. A sample is shown.

      The line «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mfrac mathcolor=¨#B94A48¨»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mfrac mathcolor=¨#B94A48¨»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/math» is coincident with «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»6«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo mathcolor=¨#B94A48¨»=«/mo»«mn mathcolor=¨#B94A48¨»0«/mn»«/math». Each line shares all of its points with the other line, so the system will have an infinite number of solutions.