Ask a New Question
Menu

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. 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 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. 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.

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.
Skip to navigation

1 answer

To determine whether the equation \( 4x - 2y = -12 \) has an infinite number of solutions, we should analyze it.

  1. Start by rewriting the equation in slope-intercept form \( y = mx + b \): \[ 4x - 2y = -12 \] Rearranging gives: \[ -2y = -4x - 12 \] Dividing by \(-2\): \[ y = 2x + 6 \] This is the equation of a line with a slope of 2 and a y-intercept of 6.

  2. An infinite number of solutions occur when you have either the same line (or multiple representations of the same line) or parallel lines, but not intersecting lines.

From the context provided, if there are multiple lines graphed:

  • If you see two parallel lines (like the solid and dotted lines in the first image), those lines will never intersect, indicating that the original line has infinite solutions shared along that parallel line.
  • If there is only one line (as in the second image), there are infinite solutions as well because any point on the line is a solution to the equation \( 4x - 2y = -12 \).
  • If there are intersecting lines (like in the last image), then there is only a single solution where they cross.

Thus, if you want to identify if the equation has an infinite number of solutions, look for evidence of parallel lines or a single line representation.

Based on your description:

  • If the first or second options represent the graph of \( 4x - 2y = -12 \), it could indicate that the equation has an infinite number of solutions.
  • If the fourth image shows intersecting lines, they share only one solution point.

Conclusion: The correct response should reflect either a scenario with one line or parallel lines for infinite solutions. If provided with specific images or further context, it can be verified which option applies accurately.