C. 2
This system of linear equations represents two lines that intersect at the point (0, 5). Therefore, there are 2 solutions to this system of equations.
Determine the number of solutions to this system of linear equations:
y = x + 5
y = −x + 5
A. infinite
B. 0
C. 2
D. 1
11 answers
Use the image to answer the question.
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The equation of the solid line is y equals negative x plus 3. The equation of the dashed line is x plus y equals 8.
Solving the system y=−x+3 and x+y=8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain.
A. The statement is false, because the lines have an intersection point.
B. The statement is true, because the lines are parallel.
C. The statement is false, because the lines are parallel.
D. The statement is true, because the lines have an intersection point.
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The equation of the solid line is y equals negative x plus 3. The equation of the dashed line is x plus y equals 8.
Solving the system y=−x+3 and x+y=8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain.
A. The statement is false, because the lines have an intersection point.
B. The statement is true, because the lines are parallel.
C. The statement is false, because the lines are parallel.
D. The statement is true, because the lines have an intersection point.
B. The statement is true, because the lines are parallel.
It is true that the lines y = -x + 3 and x + y = 8 are parallel, as they have the same slope of -1. Since parallel lines never intersect, there are no solutions to the system of equations.
It is true that the lines y = -x + 3 and x + y = 8 are parallel, as they have the same slope of -1. Since parallel lines never intersect, there are no solutions to the system of equations.
Use the image to answer the question.
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 0 comma 2 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line passes through left parenthesis negative 7 comma 0 right parenthesis and left parenthesis 7 comma 0 right parenthesis.
Does the graph show the system of equations x+y=2 and −x+y=7? Should the lines for the system be parallel?
The graph of −x + y = 7 is incorrect. The lines should intersect.
The graph of x + y = 2 is incorrect. The lines should intersect.
Both graphs are correct. The lines should be parallel.
The graph of −x + y = 7 is incorrect. The lines should be parallel.
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 0 comma 2 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The dashed line passes through left parenthesis negative 7 comma 0 right parenthesis and left parenthesis 7 comma 0 right parenthesis.
Does the graph show the system of equations x+y=2 and −x+y=7? Should the lines for the system be parallel?
The graph of −x + y = 7 is incorrect. The lines should intersect.
The graph of x + y = 2 is incorrect. The lines should intersect.
Both graphs are correct. The lines should be parallel.
The graph of −x + y = 7 is incorrect. The lines should be parallel.
The graph of −x + y = 7 is incorrect. The lines should intersect.
Since the two lines have different slopes in the system of equations x+y=2 and −x+y=7, they should intersect at one point to give the solution. Hence, the graph should show intersection, not parallel lines, accurately representing the system of equations.
Since the two lines have different slopes in the system of equations x+y=2 and −x+y=7, they should intersect at one point to give the solution. Hence, the graph should show intersection, not parallel lines, accurately representing the system of equations.
Which of these graphs shows that the linear system −2x + y = 6 and 4x − 2y = −12 has an infinite number of solutions?
A. 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.
B. 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.
C. 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.
D. 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. 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.
B. 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.
C. 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.
D. 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.
The correct option is:
B. 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 (-3, 0) and (0, 6).
This graph shows the system of equations -2x + y = 6 and 4x - 2y = -12 has an infinite number of solutions because the two equations represent the same line, hence intersecting at all points and having an infinite number of solutions.
B. 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 (-3, 0) and (0, 6).
This graph shows the system of equations -2x + y = 6 and 4x - 2y = -12 has an infinite number of solutions because the two equations represent the same line, hence intersecting at all points and having an infinite number of solutions.
Which of these graphs shows that the linear system y = −x + 6 and 3x + 3y = 18 has an infinite number of solutions?
A. 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 6 comma 0 right parenthesis.
B. 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 with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 6 comma 4 right parenthesis and left parenthesis 3 comma 3 right parenthesis.
C. 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 solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 4 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
D. 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 6 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis.
A. 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 6 comma 0 right parenthesis.
B. 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 with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 6 comma 4 right parenthesis and left parenthesis 3 comma 3 right parenthesis.
C. 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 solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 4 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
D. 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 6 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis.
C. 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 solid downward slanting line with arrows at both ends passes through the points (0, 6) and (6, 0). A dotted downward slanting line with arrows at both ends passes through the points (0, 4) and (4, 0).
This graph shows that the linear system y = −x + 6 and 3x + 3y = 18 has an infinite number of solutions because the two equations represent the same line, hence intersecting at all points and having an infinite number of solutions.
This graph shows that the linear system y = −x + 6 and 3x + 3y = 18 has an infinite number of solutions because the two equations represent the same line, hence intersecting at all points and having an infinite number of solutions.
Solve the system of equations.
x = 6y − 3
2x + 7y = 13
x = 6y − 3
2x + 7y = 13
To solve this system of equations, we will first substitute the value of x from the first equation into the second equation and then solve for y.
Given:
1) x = 6y - 3
2) 2x + 7y = 13
Substitute the value of x from equation 1 into equation 2:
2(6y - 3) + 7y = 13
12y - 6 + 7y = 13
19y - 6 = 13
19y = 19
y = 1
Now that we have found the value of y, we can substitute it back into the first equation to solve for x:
x = 6(1) - 3
x = 6 - 3
x = 3
Therefore, the solution to the system of equations is x = 3 and y = 1.
Given:
1) x = 6y - 3
2) 2x + 7y = 13
Substitute the value of x from equation 1 into equation 2:
2(6y - 3) + 7y = 13
12y - 6 + 7y = 13
19y - 6 = 13
19y = 19
y = 1
Now that we have found the value of y, we can substitute it back into the first equation to solve for x:
x = 6(1) - 3
x = 6 - 3
x = 3
Therefore, the solution to the system of equations is x = 3 and y = 1.