Statement best describe the system of equations for y = 3x + 7 x = -3x - x the equations have different slopes in different y intercepts does the system has one solution at negative 3, 7 both equations have a slope of negative 3 but they do not share the same y intercept that's the system has no solution because the lines are parallel both equations have a slope of negative 3 and they share the same wide intercept was the same system hasn't update number of solutions cuz they are identical lines the equations have different slopes but they share the same way intercept zero, three this system has one solution at 0, negative 3

Answer 1

Let's analyze the system of equations you provided step by step to determine what each statement might imply.

1. Equations:
   • The first equation is \( y = 3x + 7 \).
   • The second equation appears to have typographical errors, but if we simplify the expression \( x = -3x - x \), it simplifies to \( x = -4x \) or \( 5x = 0 \), which gives \( x = 0 \). Therefore, let’s assume the second equation is \( y = -3x \).

Now we can analyze the two equations we understood:

1. \( y = 3x + 7 \) (slope = 3, y-intercept = 7)

2. \( y = -3x \) (slope = -3, y-intercept = 0)

3. Statements Analysis:

   • "The equations have different slopes and different y-intercepts." True: The slopes are 3 and -3, and their y-intercepts are 7 and 0, respectively.
   • "Does the system have one solution at negative 3, 7?" False: The point (-3, 7) is on the first equation, but not on the second.
   • "Both equations have a slope of negative 3." False: The first has a slope of 3, the second has a slope of -3.
   • "The system has no solution because the lines are parallel." False: The slopes are different (3 and -3), so they intersect at one point.
   • "Both equations have a slope of negative 3, and they share the same y-intercept." False.
   • "The system has an infinite number of solutions because they are identical lines." False: The lines are not identical since they have different slopes and y-intercepts.
   • "The equations have different slopes but they share the same y-intercept (0, 3)." False: They do not share the same y-intercept; they intercept the y-axis at different points.
   • "This system has one solution at (0, -3)." False: At \( x = 0 \), \( y = -3 \) for the second equation, but not the first.

4. Conclusion: The correct conclusion about the system of equations is that it has one solution because the slopes are different. To find the intersection point:

Setting the equations equal: \[ 3x + 7 = -3x \]

Now solve for \( x \): \[ 6x = -7 \] \[ x = -\frac{7}{6} \]

Then substitute \( x \) into either equation to find \( y \): \[ y = 3\left(-\frac{7}{6}\right) + 7 \] \[ y = -\frac{21}{6} + \frac{42}{6} \] \[ y = \frac{21}{6} \]

So, there is one intersection point at \( \left(-\frac{7}{6}, \frac{21}{6}\right) \).

Thus, the correct statement about this system is that it has one solution, somewhere other than (0, -3) or (-3, 7).