Is 5x-10y+20 and y=-2x+6 parallel perpendicular or neither
The correct answer and explanation is:
To determine whether the lines represented by the equations 5x−10y+20=05x – 10y + 20 = 0 and y=−2x+6y = -2x + 6 are parallel, perpendicular, or neither, it is necessary to compare their slopes.
First, rewrite the equation 5x−10y+20=05x – 10y + 20 = 0 in slope-intercept form, which is y=mx+by = mx + b, where mm is the slope.
Start by isolating yy: 5x−10y+20=05x – 10y + 20 = 0 −10y=−5x−20-10y = -5x – 20 y=−5x−20−10=5×10+2010=12x+2y = \frac{-5x – 20}{-10} = \frac{5x}{10} + \frac{20}{10} = \frac{1}{2}x + 2
The slope of the first line is 12\frac{1}{2}.
The second line is already in slope-intercept form: y=−2x+6y = -2x + 6
The slope of the second line is −2-2.
Next, analyze the relationship between these slopes:
- Two lines are parallel if their slopes are equal.
- Two lines are perpendicular if the product of their slopes is −1-1.
- If neither condition is met, the lines are neither parallel nor perpendicular.
Calculate the product of the slopes: (12)×(−2)=−1\left(\frac{1}{2}\right) \times (-2) = -1
Since the product of the slopes is −1-1, the two lines are perpendicular.
Summary:
The first line has slope 12\frac{1}{2}, and the second line has slope −2-2. The product of the slopes is −1-1, which means the lines are perpendicular. Lines with slopes that multiply to −1-1 always intersect at a right angle. Therefore, these two lines are perpendicular to each other.