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Systems of Linear Equations Unit Test

6 of 146 of 14 Items

Question
Which of these graphs shows that the linear system 3x+2y=12
and 6x+4y=24
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 intersecting lines are plotted on the plane. A solid downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 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 downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 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. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 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. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 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 parallel lines are plotted on the plane. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 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. Two parallel lines are plotted on the plane. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 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. A downward slanting line with arrows at both ends passes through the points left parenthesis negative 10 comma 9 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis.

1st 2nd 3rd or 4th option

1 answer

To determine which graph shows that the linear system of equations \(3x + 2y = 12\) and \(6x + 4y = 24\) has an infinite number of solutions, we should first analyze the equations.

Notice that the second equation, \(6x + 4y = 24\), can be simplified by dividing all terms by 2, resulting in: \[ 3x + 2y = 12 \]

Since both equations represent the same line, this indicates that they have an infinite number of solutions (any point on the line is a solution to both equations).

Now, looking at the options you provided:

  1. First option: Two intersecting lines - this would indicate a single solution (they intersect at one point).
  2. Second option: One line - this represents a single equation and does not show multiple solutions.
  3. Third option: Two parallel lines - this means no solutions, as parallel lines never intersect.
  4. Fourth option: One line - this again represents only one equation, not showing the infinite solutions of the two equations being the same.

Thus, the correct answer is none of the options perfectly show that there are infinite solutions since they all indicate different scenarios. The correct graph for infinite solutions should show one line representing both equations.