(1 point) Responses A system of equations is graphed on a coordinate plane. One line passes through points left parenthesis 0 comma 2 right parenthesis and left parenthesis negative 2 comma negative 4 right parenthesis. The other line passes through left parenthesis negative 2 comma 7 right parenthesis and left parenthesis 2 comma 9 right parenthesis. The lines intersect at labeled point left parenthesis negative 4 comma 6 right parenthesis. Image with alt text: A system of equations is graphed on a coordinate plane. A system of equations is graphed on a coordinate plane. One line passes through points left parenthesis 0 comma 2 right parenthesis and left parenthesis negative 2 comma negative 4 right parenthesis. The other line passes through left parenthesis negative 2 comma 7 right parenthesis and left parenthesis 2 comma 9 right parenthesis. The lines intersect at labeled point left parenthesis negative 4 comma 6 right parenthesis. A system of equations is graphed on a coordinate plane. One line passes through points left parenthesis 0 comma 2 right parenthesis and left parenthesis 2 comma 4 right parenthesis. The other line passes through left parenthesis negative 2 comma 9 right parenthesis and left parenthesis 2 comma 7 right parenthesis. The lines intersect at labeled point left parenthesis 4 comma 6 right parenthesis. Image with alt text: A system of equations is graphed on a coordinate plane. A system of equations is graphed on a coordinate plane. One line passes through points left parenthesis 0 comma 2 right parenthesis and left parenthesis 2 comma 4 right parenthesis. The other line passes through left parenthesis negative 2 comma 9 right parenthesis and left parenthesis 2 comma 7 right parenthesis. The lines intersect at labeled point left parenthesis 4 comma 6 right parenthesis. A system of equations is graphed on a coordinate plane. One line passes through points left parenthesis negative 2 comma 4 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The other line passes through left parenthesis negative 2 comma 1 right parenthesis and left parenthesis 2 comma 3 right parenthesis. The lines intersect at labeled point left parenthesis 0 comma 2 right parenthesis. Image with alt text: A system of equations is graphed on a coordinate plane. A system of equations is graphed on a coordinate plane. One line passes through points left parenthesis negative 2 comma 4 right parenthesis and left parenthesis 2 comma 0 right parenthesis. The other line passes through left parenthesis negative 2 comma 1 right parenthesis and left parenthesis 2 comma 3 right parenthesis. The lines intersect at labeled point left parenthesis 0 comma 2 right parenthesis.
The correct response is:
A system of equations is graphed on a coordinate plane. One line passes through points (0, 2) and (-2, -4). The other line passes through (-2, 7) and (2, 9). The lines intersect at labeled point (-4, 6).
Based on the given information, the system of equations can be represented as follows:
Equation 1:
Passes through the points (0, 2) and (-2, -4).
Equation 2:
Passes through the points (-2, 7) and (2, 9).
The lines intersect at the labeled point (-4, 6).
To determine which set of points represents the correct graph of the system of equations, we can analyze the information provided.
The first line passes through the points (0, 2) and (-2, -4), and the second line passes through the points (-2, 7) and (2, 9). The lines intersect at the point (-4, 6).
To verify if the given set of points matches the given information, we can check if the points satisfy the equations of the lines.
For Line 1:
- When substituting the point (0, 2) into the equation, it should be true.
- When substituting the point (-2, -4) into the equation, it should also be true.
For Line 2:
- When substituting the point (-2, 7) into the equation, it should be true.
- When substituting the point (2, 9) into the equation, it should also be true.
Let's calculate the equations for both lines and check if the given points satisfy them.
Line 1 equation:
Using the point-slope form, the equation for Line 1 can be found by taking the slope between (0, 2) and (-2, -4):
slope = (y2 - y1) / (x2 - x1)
= (-4 - 2) / (-2 - 0)
= -6 / -2
= 3
Using the point-slope form with the slope (3) and the point (0, 2):
y - y1 = m(x - x1)
y - 2 = 3(x - 0)
y - 2 = 3x
y = 3x + 2
Now, we can substitute the given points into this equation.
For (0, 2):
2 = 3(0) + 2
2 = 0 + 2
2 = 2 (true)
For (-2, -4):
-4 = 3(-2) + 2
-4 = -6 + 2
-4 = -4 (true)
The points (0, 2) and (-2, -4) satisfy the equation y = 3x + 2.
Line 2 equation:
Using the point-slope form, the equation for Line 2 can be found by taking the slope between (-2, 7) and (2, 9):
slope = (y2 - y1) / (x2 - x1)
= (9 - 7) / (2 - (-2))
= 2 / 4
= 1/2
Using the point-slope form with the slope (1/2) and the point (-2, 7):
y - y1 = m(x - x1)
y - 7 = (1/2)(x - (-2))
y - 7 = (1/2)(x + 2)
y - 7 = (1/2)x + 1
y = (1/2)x + 8
Now, we can substitute the given points into this equation.
For (-2, 7):
7 = (1/2)(-2) + 8
7 = -1 + 8
7 = 7 (true)
For (2, 9):
9 = (1/2)(2) + 8
9 = 1 + 8
9 = 9 (true)
The points (-2, 7) and (2, 9) satisfy the equation y = (1/2)x + 8.
Based on these calculations, the set of points (-4, 6) is the correct intersection point for the given system of equations.