To determine which equations represent a line that is perpendicular to -x + 2y = 4 and passes through (-2, 1), we need to find the equation(s) with a slope that is the negative reciprocal of the slope of -x + 2y = 4.
The given equation is in standard form Ax + By = C, where A, B, and C are constants.
To find the slope of the given line, we need to rewrite it in slope-intercept form y = mx + b, where m represents the slope.
-x + 2y = 4
2y = x + 4
y = (1/2)x + 2
The slope of the given line is 1/2. The negative reciprocal of 1/2 is -2.
Now let's analyze each equation:
y = -2x + 5: The slope of this equation is -2, which is the negative reciprocal of the slope of -x + 2y = 4. Therefore, y = -2x + 5 is perpendicular to -x + 2y = 4.
2x + y = -3: To determine the slope of this equation, we need to rearrange it into slope-intercept form y = mx + b. Subtracting 2x from both sides, we get y = -2x - 3, which has a slope of -2. The slope of this equation (-2) is the same as the slope of -x + 2y = 4, not its negative reciprocal. Therefore, 2x + y = -3 is not perpendicular to -x + 2y = 4.
y = -2x - 3: The slope of this equation is -2, which is the negative reciprocal of the slope of -x + 2y = 4. Therefore, y = -2x - 3 is perpendicular to -x + 2y = 4.
|-2x + y = 5: Rearranging this equation into slope-intercept form, we get y = 2x + 5, which has a slope of 2. The slope of this equation (2) is not the negative reciprocal of the slope of -x + 2y = 4. Therefore, |-2x + y = 5 is not perpendicular to -x + 2y = 4.
Based on the analysis, the correct equations representing a line that is perpendicular to -x + 2y = 4 and passes through (-2, 1) are:
- y = -2x + 5
- y = -2x - 3