On a coordinate plane, 2 dashed lines are shown. The first straight line has a negative slope and goes through (0, 1) and (2, 0). Everything below and to the left of the line is shaded. The second straight line has a positive slope and goes through (negative 1, 0) and (0, 2). Everything above and to the left of the line is shaded. A point is shown at (negative 3, 1).
Which system of inequalities with a solution point is represented by the graph?
y > 2x – 2 and y < Negative one-halfx – 1; (3, 1)
y > 2x – 2 and y < Negative one-halfx + 1; (–3, 1)
y > 2x + 2 and y < Negative one-halfx – 1; (3, 1)
y > 2x + 2 and y < Negative one-halfx + 1; (–3, 1)
help
wheres the answer
To determine which system of inequalities is represented by the graph, we need to analyze the given information.
First, let's consider the first straight line. Given that it has a negative slope and passes through the points (0, 1) and (2, 0), we can find the slope using the slope formula:
slope = (y2 - y1) / (x2 - x1) = (0 - 1) / (2 - 0) = -1/2
So, the equation of the first line is y = -1/2x + b. To find the y-intercept (b), we can substitute the coordinates of one of the given points, such as (0, 1), into the equation:
1 = -1/2(0) + b
1 = b
Therefore, the equation of the first line is y = -1/2x + 1.
Next, let's consider the second straight line. Given that it has a positive slope and passes through the points (-1, 0) and (0, 2), we can find the slope using the same formula:
slope = (y2 - y1) / (x2 - x1) = (2 - 0) / (0 - (-1)) = 2/1 = 2
So, the equation of the second line is y = 2x + b. Substituting the coordinates of one of the given points, such as (-1, 0), into the equation:
0 = 2(-1) + b
0 = -2 + b
b = 2
Therefore, the equation of the second line is y = 2x + 2.
Now, let's analyze the shading. The first line indicates that everything below and to the left of the line is shaded. The second line indicates that everything above and to the left of the line is shaded.
Given that the point (-3, 1) is plotted, we can substitute its coordinates into the equations of the lines to check if the inequalities hold true.
Considering the first line,
1 > 2(-3) - 2
1 > -6 - 2
1 > -8, which is true.
Considering the second line,
1 > 2(-3) + 2
1 > -6 + 2
1 > -4, which is also true.
Based on this analysis, the correct system of inequalities with a solution point represented by the graph is:
y > 2x - 2 and y < -1/2x + 1; (-3, 1)
Therefore, the correct answer is:
y > 2x - 2 and y < -1/2x + 1; (-3, 1)