Select the number of solutions for each system of equations.

Question

Response area with 12 radio buttons within 4 groups.
Equation Number of Solutions
 
System of Equations 1st Row y equals 2 x plus 5 2nd Row y equals 2 x plus 10
 
 none  one  infinite 
 
System of Equations 1st Row y equals 2 x plus 5 2nd Row y equals 2 x plus 5
 
 none  one  infinite 
 
System of Equations 1st Row y equals 2 x plus 5 2nd Row y equals 2 x minus 5
 
 none  one  infinite 
 
System of Equations 1st Row y equals 2 x plus 5 2nd Row y equals negative 2 x plus 5
 
 none  one  infinite 

Answer 1

To determine the number of solutions for each system of equations, we analyze them one by one:

1. System of Equations:

   • \( y = 2x + 5 \)
   • \( y = 2x + 10 \)

   Here, the slopes are the same (both are 2), but the y-intercepts are different (5 and 10). This indicates that the lines are parallel and will never intersect, resulting in no solutions.

2. System of Equations:

   • \( y = 2x + 5 \)
   • \( y = 2x + 5 \)

   In this case, both equations are identical. Thus, all points on the line are solutions. This means there are infinitely many solutions.

3. System of Equations:

   • \( y = 2x + 5 \)
   • \( y = 2x - 5 \)

   Again, the slopes are the same (both are 2) but the y-intercepts are different (5 and -5). This indicates that the lines are parallel and will never intersect, resulting in no solutions.

4. System of Equations:

   • \( y = 2x + 5 \)
   • \( y = -2x + 5 \)

   Here, the slopes are different (2 and -2), so the lines will intersect at one point. Therefore, there is one solution.

Summary:

• First system: none
• Second system: infinite
• Third system: none
• Fourth system: one

You can select the following radio buttons for your response:

• First system: none
• Second system: infinite
• Third system: none
• Fourth system: one