I have a test tomorrow with a few sample questions that are expected to be on the test that I do not know how to properly do.
1)Find the slope intercept equation of the line passing through the intersection of the lines 2x-4y=1 & 3x+4y=4 and parallel to the line 5x+7y+3=0
2)Find the angle between the lines described by y=3x-5 & 4x+2y=12
3) Find the general form of the equation of the line through the intersection of 5x-2y=2 & 2x-3y=3 and perpendicular to the line through (4,-1) and (1,-6)
4)Calculate the vertex, x intercept & y intercept. State if the vertex represents a maximum or minimum and opening upwards or downwards. y=-3x^2 + 6x + 9
5a) Determine the equation of the line parallel to the x axis and passes through (8,-14)
b)Find the slope-intercept form equation between two points (-8,14) and (-7,-3)
c)Calculate the distance between two points
d) Calculate midpoint between the two points
Thanks in advance!
#1
I assume you can find the intersection of two lines. In this case, that is at (1,1/4).
The line 5x+7y+3=0 has slope -5/7, so now we have a point and a slope, and the desired solution is
y - 1/4 = -5/7 (x-1)
#2
y=3x-5 has slope 3, so the angle it makes with the x-axis is arctan(3)
4x+2y=12 has slope -1/2, so it makes an angle of arctan(-1/2)
The angle between the lines is thus arctan(3)-arctan(-1/2). That will give you an obtuse angle. There is also an acute angle, which is its supplement.
#3 is just like #1, except the new line's slope is the negative reciprocal of the slope between the two points.
#4
y=-3x^2 + 6x + 9
You know it opens downward because of the negative coefficient for x^2
Now, to get the vertex, complete the square
y = -3(x^2-2x+3) = -3(x-3)(x+1)
Now you can read off the x-intercepts, and the y-intercept (from the original equation) is y=9.
#5
(a) naturally, the line is y = -14
(b) just use the two-point form:
y-14 = (-7-14)/(-3+8) (x+8)
and massage that.
(c) I assume you know the distance formula
(d) the midpoint is just the average of the two points: ((-8 + -7)/2 , (14 + -3)/2)
1) To find the slope-intercept equation of a line passing through the intersection of two given lines and parallel to a third line, follow these steps:
Step 1: Find the intersection point of the two given lines.
- Set the two equations equal to each other: 2x - 4y = 1 and 3x + 4y = 4.
- Solve the system of equations to find the values of x and y that satisfy both equations. This will give you the intersection point of the lines.
Step 2: Determine the slope of the third line.
- The slope-intercept form of a line is y = mx + b, where m is the slope.
- The given equation is 5x + 7y + 3 = 0. Rewrite it in slope-intercept form by isolating y.
Step 3: Use the intersection point and the slope to find the equation of the line.
- The equation of the line passing through the intersection point and parallel to the third line will have the same slope as the third line.
- Plug in the slope and the coordinates of the intersection point into the slope-intercept form (y = mx + b) to find the equation.
2) To find the angle between two lines described by equations, follow these steps:
Step 1: Write the equations of the lines in slope-intercept form (y = mx + b).
Step 2: Determine the slopes of the lines by comparing the coefficients of x in the equation.
Step 3: Use the formula for the angle between two lines:
- The formula is tan(θ) = |(m1 - m2) / (1 + m1 * m2)|, where m1 and m2 are the slopes of the two lines.
3) To find the general form of the equation of a line passing through the intersection of two given lines and perpendicular to a third line, follow these steps:
Step 1: Find the intersection point of the two given lines by solving the system of equations.
Step 2: Determine the slope of the third line.
Step 3: Find the negative reciprocal of the third line's slope (denoted as -1/m).
Step 4: Use the intersection point and the negative reciprocal slope to write the equation in point-slope form (y - y1 = -1/m * (x - x1)).
Step 5: Simplify the equation by rearranging the terms and converting it to the general form (Ax + By + C = 0).
4) To calculate the vertex, x-intercept, y-intercept, and determine if the vertex represents a maximum or minimum and opening upwards or downwards for a quadratic equation in the form y = ax^2 + bx + c, follow these steps:
Step 1: Find the vertex using the formula x = -b / 2a.
- Substitute the values of a, b, and c into the formula to find the x-coordinate of the vertex.
Step 2: Substitute the x-coordinate of the vertex back into the original equation to find the y-coordinate of the vertex.
Step 3: To find the x-intercepts, set the equation equal to zero and solve for x. These are the points where the graph intersects the x-axis.
Step 4: To find the y-intercept, substitute x = 0 into the equation and solve for y. This is the point where the graph intersects the y-axis.
Step 5: Determine if the vertex represents a maximum or minimum and whether the graph opens upwards or downwards.
- If the coefficient of the x^2 term (a) is positive, the graph opens upwards, and the vertex represents a minimum.
- If the coefficient of the x^2 term (a) is negative, the graph opens downwards, and the vertex represents a maximum.
5a) The equation of a line parallel to the x-axis is y = c, where c is a constant representing the y-coordinate. Since the line passes through (8, -14), the equation is y = -14.
b) To find the slope-intercept form equation between two points (-8, 14) and (-7, -3), follow these steps:
- Find the slope (m) using the formula m = (y2 - y1) / (x2 - x1).
- Choose one of the given points and substitute its coordinates and the slope into the slope-intercept form equation (y = mx + b) to solve for b.
- Write the equation using the found slope (m) and the solved value of b.
c) To calculate the distance between two points (x1, y1) and (x2, y2), apply the distance formula:
- The distance formula is d = √((x2 - x1)^2 + (y2 - y1)^2).
- Substitute the given coordinates into the formula and solve for d.
d) To find the midpoint between two points (x1, y1) and (x2, y2), apply the midpoint formula:
- The midpoint formula is ( (x1 + x2) / 2 , (y1 + y2) / 2 ).
- Substitute the given coordinates into the formula to find the midpoint.