n the same graph, what is the approximate y-value of the point of intersection? A solid line passes through the points left-parenthesis 0 comma 4 right-parenthesis and left-parenthesis 2 comma 8 right-parenthesis. The region above the line is shaded red.%C2%A0A dotted line passes through the points left-parenthesis 0 comma 2 right-parenthesis and left-parenthesis 2 comma 0 right-parenthesis. The region below the line is shaded blue. A region of the coordinate plane is shaded both red and blue. (1 point) Responses –1 –1 4 4 3 3 2
To find the y-value of the point of intersection, we can set the two equations equal to each other and solve for y.
The equation of the solid line passing through the points (0, 4) and (2, 8) can be written as:
y = mx + b
where m is the slope and b is the y-intercept. To find the slope, we use the formula:
m = (y2 - y1) / (x2 - x1)
Plugging in the coordinates (0, 4) and (2, 8):
m = (8 - 4) / (2 - 0)
m = 4 / 2
m = 2
Now we can find the value of b by plugging in the coordinates (0, 4) and the slope 2:
4 = 2(0) + b
4 = b
So the equation of the solid line is:
y = 2x + 4
Similarly, the equation of the dotted line passing through the points (0, 2) and (2, 0) can be calculated. The slope is:
m = (0 - 2) / (2 - 0)
m = -2 / 2
m = -1
And using the coordinates (0, 2) and slope -1:
2 = -1(0) + b
2 = b
So the equation of the dotted line is:
y = -x + 2
To find the point of intersection, we set the two equations equal to each other:
2x + 4 = -x + 2
Combining like terms:
3x + 4 = 2
Subtracting 4 from both sides:
3x = -2
Dividing by 3:
x = -2/3
Plugging this value of x back into either equation, we can find the corresponding y-value:
y = 2(-2/3) + 4
y = -4/3 + 4
y = -4/3 + 12/3
y = 8/3
So the approximate y-value of the point of intersection is 8/3, which is approximately 2.6667. Therefore, the answer is 3.