To find the equation of a line that is perpendicular to the line represented by 2x + y = 7, we first need to determine the slope of the given line.
We can rewrite the equation in slope-intercept form (y = mx + b), where m represents the slope. Starting from the original equation:
2x + y = 7
We can isolate y:
y = -2x + 7
From this form, we see that the slope (m) of the line is -2.
When two lines are perpendicular, the slopes are negative reciprocals of each other. Therefore, the slope of the line that is perpendicular to our given line will be:
mperpendicular = -1/(-2) = 1/2
Now that we have the slope of the perpendicular line, we can use the point-slope form of the equation of a line, which is given by:
y – y1 = m(x – x1)
Where (x1, y1) is a point on the line. If we choose any point (for example, (0, 7) which lies on the original line), we can plug it into the point-slope formula:
y – 7 = (1/2)(x – 0)
Simplifying this gives us:
y – 7 = (1/2)x
y = (1/2)x + 7
Thus, one possible equation of the line that is perpendicular to the line 2x + y = 7 is:
y = (1/2)x + 7
This indicates that the new line, with slope 1/2, intersects the y-axis at 7.