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The graphs represent this system:

y = 3х - 2
-3x + y=7
How many solutions does this system have: 0, 1, or infinite
(1 point)

7 answers

To determine how many solutions the system of equations has, we can analyze the given equations:

  1. First Equation: \( y = 3x - 2 \)

    • This is in slope-intercept form \( y = mx + b \), where the slope \( m = 3 \) and the y-intercept \( b = -2 \).

  2. Second Equation: \( -3x + y = 7 \)

    • We can rearrange this equation into slope-intercept form: \[ y = 3x + 7 \]
    • This has a slope \( m = 3 \) and a y-intercept \( b = 7 \).

Now let's analyze the slopes and y-intercepts:

  • Both lines have the same slope of \( 3 \).
  • The y-intercepts are different (\(-2\) for the first equation and \(7\) for the second equation).

Since both lines are parallel (have the same slope) but have different y-intercepts, they will never intersect.

Thus, the system of equations has 0 solutions.

If the lines in a system of linear equations are parallel, what is the sysi called? Enter the number that corresponds to the correct option:
1 - consistent system of equations
2 - dependent system of equations
3 - inconsistent system of equations
4 - independent system of equations
(1 point)

If the lines in a system of linear equations are parallel, the system is called an inconsistent system of equations. Therefore, the correct option is:

3 - inconsistent system of equations.

Solve the given system of linear equations by inspection. Enter 1 for one solution, 2 for no solution, or 3 for infinitely many solutions.
2x - 7y=4
2х -7y=-4
(1 point)

To analyze the given system of linear equations:

  1. The first equation is \( 2x - 7y = 4 \).
  2. The second equation is \( 2x - 7y = -4 \).

Now, let's observe the structure of these equations:

  • The left-hand sides of both equations are identical: \( 2x - 7y \).
  • However, the right-hand sides are different: \( 4 \) in the first equation and \( -4 \) in the second equation.

Since these two equations represent two parallel lines (they have the same slopes and different y-intercepts), they will never intersect. Therefore, there are no solutions to this system.

Thus, the answer is:

2 for no solution.

Solve the given system of linear equations by inspection. Enter 1 for one solution, 2 for no solution, or 3 for infinitely many solutions.
У = - 3х + 5
2y = - 6x + 10
(1 point)

To analyze the given system of linear equations:

  1. The first equation is \( y = -3x + 5 \).
  2. The second equation is \( 2y = -6x + 10 \).

Now, let's simplify the second equation. We can divide the entire equation by \( 2 \):

\[ y = -3x + 5. \]

Now we have both equations in the same form:

  1. \( y = -3x + 5 \)
  2. \( y = -3x + 5 \)

Since both equations are identical, they represent the same line. This means that there are infinitely many solutions because every point on the line is a solution to the system of equations.

Thus, the answer is:

3 for infinitely many solutions.