which graphs with no solution correctly shows the system of equations y = 2x + 4 and -2x + y = 8?

Answer 1

To determine whether the system of equations \( y = 2x + 4 \) and \( -2x + y = 8 \) has a solution, we first need to analyze the two equations.

1. The first equation is already in slope-intercept form: \[ y = 2x + 4 \] This line has a slope of 2 and a y-intercept of 4.

2. The second equation can be rearranged into slope-intercept form: \[ -2x + y = 8 \implies y = 2x + 8 \] This line also has a slope of 2 but a y-intercept of 8.

Now, we can see that both lines have the same slope (2) but different y-intercepts (4 and 8). This means that the lines are parallel and will never intersect.

Since parallel lines do not intersect, there is no solution to this system of equations.

If you were to graph these lines, you would plot both lines, and they will run parallel to each other without touching at any point. A graph of this situation would clearly illustrate that there is no solution, as it visually demonstrates the two parallel lines.

In summary, to correctly show the system of equations \( y = 2x + 4 \) and \( -2x + y = 8 \) has no solution, you would represent two parallel lines on the graph.

Answer 2

Which of these graphs shows that the linear system 3x + 2y = 12 and 6x + 4y = 24 has an infinite number of solutions?

Answer 3

To determine whether the system of equations \(3x + 2y = 12\) and \(6x + 4y = 24\) has an infinite number of solutions, we can analyze the equations:

1. The first equation is: \[ 3x + 2y = 12 \] We can rearrange it into slope-intercept form (y = mx + b): \[ 2y = -3x + 12 \implies y = -\frac{3}{2}x + 6 \]

2. The second equation is: \[ 6x + 4y = 24 \] We can also rearrange this into slope-intercept form: \[ 4y = -6x + 24 \implies y = -\frac{3}{2}x + 6 \]

Now, if we compare the two equations, we can see that they are equivalent. The second equation is simply a multiple of the first equation:

\[ 6x + 4y = 2(3x + 2y = 12) \]

Since both equations represent the same line in the Cartesian plane, they will overlap entirely.

This means the system has an infinite number of solutions because every point on that line is a solution to the system.

To represent this on a graph, you would display one line, as drawing both equations would show them as the same line. Therefore, you are looking for a graph that exhibits a single line, indicating that the two equations are the same and hence that there are infinitely many solutions.