Ask a New Question
Menu

Which of these graphs shows that the linear system −2x+y=6

and 4x−2y=−12
has an infinite number of solutions?(1 point)
Responses

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. An upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. An upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 1 comma 7 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 1 comma 7 right parenthesis.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A solid upward slanting line with arrows at both ends, parallel to the dotted line, passes through origin.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A solid upward slanting line with arrows at both ends, parallel to the dotted line, passes through origin.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. An upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 3 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. An upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 3 comma 0 right parenthesis.
Skip to navigation

1 answer

To determine which graph shows that the linear system has an infinite number of solutions, we need to check if the two equations represent the same line.

The given equations are:

  1. \( -2x + y = 6 \)
  2. \( 4x - 2y = -12 \)

Let's rewrite the second equation in a more comparable form:

  1. Multiply both sides of the second equation by \( -1 \): \[ -4x + 2y = 12 \]

  2. Now, we can rearrange it to compare with the first equation. Divide by 2: \[ 2y - 4x = 12 \] or \[ y = 2x + 6 \]

  3. The first equation can also be rearranged: \[ y = 2x + 6 \]

Since both equations simplify to the same line \( y = 2x + 6 \), the graphs of these two equations will overlap completely. This indicates that the system has infinite solutions because any point on this line is a solution to both equations.

Conclusion:

The correct graph showing that the system has an infinite number of solutions will depict two overlapping lines (i.e., one line on top of another), which means they represent the same line.

Among your options, look for the graph that features two lines that coincide or are the same, which indicates that there are infinitely many solutions. The option showing two parallel lines would indicate no solution, while the option with two intersecting lines would indicate a unique solution.

If neither of the responses provided describes two coinciding lines, then that is the required answer. Please review your descriptions for that specific depiction.