Tell whether the lines for each pair of equations are parallel, perpendicular, or neither.
y = –4x + 3
–2x + 8y = 5
To determine the relationship between the lines represented by the pair of equations, we can compare the slopes (coefficients of x) of the equations.
The first equation is in slope-intercept form, y = mx + b, where m is the slope of the line. In this case, the slope is -4.
The second equation can be rearranged to slope-intercept form by isolating y: -2x + 8y = 5 -> 8y = 2x + 5 -> y = (2/8)x + 5/8. Thus, the slope of the second line is 2/8 = 1/4.
Since the slopes of the lines are different (-4 and 1/4), the lines are not parallel. To determine if they are perpendicular, we can check if the product of their slopes is -1.
-4 * (1/4) = -1.
Since the product is -1, the lines are perpendicular to each other.
Bot Tell whether the lines for each pair of equations are parallel, perpendicular, or neither.
y = (negative one fourth)x + 8
–2x + 8y = 4
To determine the relationship between the lines represented by the pair of equations, we can compare the slopes (coefficients of x) of the equations.
The first equation is in slope-intercept form, y = mx + b, where m is the slope of the line. In this case, the slope is -1/4.
The second equation can be rearranged to slope-intercept form by isolating y: -2x + 8y = 4 -> 8y = 2x + 4 -> y = (2/8)x + 4/8. Thus, the slope of the second line is 2/8 = 1/4.
Since the slopes of the lines are the same (-1/4 and 1/4), the lines are parallel to each other.
bot Write the equation of a line that is perpendicular to the given line and that passes through the given point.
y – 3 = eight-thirds(x + 2); (–2, 3)
To find the equation of a line that is perpendicular to the given line and passes through the given point, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
The given line is in slope-intercept form, y - 3 = (8/3)(x + 2). By rearranging the equation, we can determine its slope:
y - 3 = (8/3)(x + 2)
y - 3 = (8/3)x + (16/3)
y = (8/3)x + (16/3) + 3
y = (8/3)x + (16/3) + 9/3
y = (8/3)x + (25/3)
From the given equation, we can see that the slope is 8/3.
To find the slope of the perpendicular line, we need to determine the negative reciprocal of 8/3. To do this, we flip the fraction and change the sign. Therefore, the slope of the perpendicular line is -3/8.
Now that we have the slope (-3/8) and the given point (-2, 3), we can use the point-slope form of a line to find the equation:
y - y1 = m(x - x1)
y - 3 = (-3/8)(x - (-2))
y - 3 = (-3/8)(x + 2)
y - 3 = (-3/8)x - 3/4
y = (-3/8)x - 3/4 + 3
y = (-3/8)x - 3/4 + 12/4
y = (-3/8)x + 9/4
Therefore, the equation of the line that is perpendicular to the given line and passes through the point (-2, 3) is y = (-3/8)x + 9/4.
To determine whether two lines are parallel, perpendicular, or neither, we need to compare their slopes.
The given equations are:
1) y = -4x + 3
2) -2x + 8y = 5
To identify the slope of the first equation, we can note that it follows the standard form y = mx + b, where m represents the slope. In this case, the slope is -4.
The second equation is not in slope-intercept form, so we need to rearrange it. Subtracting -2x from both sides and dividing by 8, we get:
8y = 2x + 5
y = (2/8)x + (5/8)
Now we can compare the slopes. The slope of the second equation is 2/8, which simplifies to 1/4.
Since the slopes of the two equations are different (-4 and 1/4), the lines are neither parallel nor perpendicular.