Which expression is equivalent to 4x2−(8x−4x2)+5+(4x−10)?
2 answers:
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Answer:
a. The equation of the parallel line to the given line is y = -4x + 19
b. The equation of the perpendicular line to the given line is y = x + 2
Step-by-step explanation:
Parallel lines have the same slopes
- If the slope of one of them is m, then the slope of the other is m
The product of the slopes of the perpendicular lines is -1
- If the slope of one of them is m, then the slope of the other is
- To find the slope of a perpendicular line to a given line reciprocal the slope of the given line and change its sign
The rule of the slope is m = , where
- (x1, y1) and (x2, y2) are the points on the line
The form of the equation of a line is y = m x + b, where
- m is the slope
- b is the y-intercept
Let us solve the question
∵ The given line passes through points (1, 6) and (2, 2)
∴ x1 = 1 and y1 = 6
∴ x2 = 2 and y2 = 2
→ Substitute them in the rule of the slope to find it
∵ m =
∴ The slope of the given line is -4
a.
∵ The line is parallel to the given line
∴ Their slopes are equal
∵ The slope of the given line = -4
∴ The slope of the parallel line = -4
→ Substitute its value in the form of the equation above
∴ y = -4x + b
→ To find b substitute x and y in the equation by the coordinates
of any point on the line
∵ The parallel line passes through the point (4, 3)
∴ x = 4 and y = 3
∵ 3 = -4(4) + b
∴ 3 = -16 + b
→ Add 16 to both sides
∴ 3 + 16 = -16 + 16 + b
∴ 19 = b
→ Substitute it in the equation
∴ y = -4x + 19
The equation of the parallel line to the given line is y = -4x + 19
b.
∵ The line is perpendicular to the given line
∴ The product of their slopes is -1
→ Reciprocal the slope of the given line and change its sign
∵ The slope of the given line = -4
∴ The slope of the perpendicular line =
→ Substitute its value in the form of the equation above
∴ y = x + b
→ To find b substitute x and y in the equation by the coordinates
of any point on the line
∵ The perpendicular line passes through the point (4, 3)
∴ x = 4 and y = 3
∵ 3 = (4) + b
∴ 3 = 1 + b
→ Subtract 1 from both sides
∴ 3 - 1 = 1 - 1 + b
∴ 2 = b
→ Substitute it in the equation
∴ y = x + 2
The equation of the perpendicular line to the given line is y = x + 2
With convolution theorem the equation is proved.
According to the statement
we have given that the equation and we have to evaluate with the convolution theorem.
Then for this purpose, we know that the
A convolution integral is an integral that expresses the amount of overlap of one function as it is shifted over another function.
And the given equation is solved with this given integral.
So, According to this theorem the equation becomes the
Then after solving, it become and with theorem it says that the
Hence by this way the given equation with convolution theorem is proved.
So, With convolution theorem the equation is proved.
Learn more about convolution theorem here
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