solve the following system of equations with your choice of number in the box.
3x+2y=7
-3x-2y= (box)
how many solutions did you find?
pick a different number in the box that results in a different number of solutions
Let's solve the system of equations:
1) 3x + 2y = 7
2) -3x - 2y = (box)
Choosing a number in the box:
If we pick any number that is not equal to 7, we will find that the system of equations has a unique solution.
Let's choose the number 6 in the box.
Substituting the value into equation 2:
-3x - 2y = 6
Now, we can add equation 1 and equation 2:
(3x + 2y) + (-3x - 2y) = 7 + 6
0x + 0y = 13
0 = 13
This equation is not true, which means the system of equations is inconsistent. Therefore, the system of equations has no solution.
Different number in the box:
If we pick the number 7 in the box, we can see that the system of equations becomes:
-3x - 2y = 7
Adding equation 1 and equation 2:
(3x + 2y) + (-3x - 2y) = 7 + 7
0x + 0y = 14
0 = 14
Again, this equation is not true, meaning the system of equations is inconsistent. Therefore, the system of equations has no solution.
In both cases, we find that the system of equations has no solution.
To solve the system of equations with the chosen number, we will use the method of elimination.
1. Start by adding the two equations together. The x-terms will cancel out.
(3x + 2y) + (-3x - 2y) = 7 + (box)
0x + 0y = 7 + (box)
0 = 7 + (box)
2. The resulting equation is 0 = 7 + (box), which can be rewritten as 0 = (box + 7).
Now let's consider the different scenarios based on the number we choose for the box:
1. If we choose a number that makes the equation 0 = (box + 7) true, such as box = -7, then the system of equations becomes:
0 = (-7 + 7)
0 = 0
In this case, the system has infinitely many solutions since the two equations represent the same line.
2. If we choose a number that makes the equation 0 = (box + 7) false, such as box = 1, then the system of equations becomes:
0 = (1 + 7)
0 = 8
In this case, the system has no solutions since the resulting equation is not true.
Therefore, depending on the number chosen for the box, we can find either infinitely many solutions or no solutions for the system of equations.
To solve the given system of equations, we can use the method of elimination. Adding the two equations will cancel out the x and y terms, as they have opposite coefficients:
(3x + 2y) + (-3x - 2y) = 7 + (box)
Simplifying this equation, we get:
0 = 7 + (box)
Since we want to find the value of (box) that results in a different number of solutions, we need to make the equation invalid. In this case, any value of (box) other than -7 will make the equation invalid and result in no solution.
So, if we choose any value other than -7 for (box), the system will have no solution. For example, let's say we choose (box) = 5:
0 = 7 + 5
This is not a true statement, so the system has no solution.
On the other hand, if we choose (box) = -7, then the equation becomes:
0 = 7 + (-7)
This simplifies to 0 = 0, which is a true statement. In this case, the system has infinitely many solutions, as the two equations represent the same line.
Therefore, by choosing (box) = -7, we found infinitely many solutions. By choosing any other value for (box), we found no solution.