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) To find the slope-intercept equation of a line passing through the intersection of two lines and parallel to another line, you can follow these steps:
Step 1: Find the point of intersection of the two given lines.
- Solve the system of equations formed by the two given lines to find the values of x and y at their intersection point.
Step 2: Find the slope of the given line.
- The slope-intercept form of a line is y = mx + b, where m represents the slope of the line.
Step 3: Use the point of intersection and the slope to write the equation.
- Substitute the values of the point of intersection and the slope into the slope-intercept form equation to get the final answer.
2) To find the angle between two lines, you can use the formula:
angle = arctan |(m1 - m2) / (1 + m1 * m2)|
where m1 and m2 are the slopes of the lines.
3) To find the general form of the equation of a line passing through the intersection of two given lines and perpendicular to another line passing through two given points, follow these steps:
Step 1: Find the point of intersection of the two given lines.
- Solve the system of equations formed by the two given lines to find the values of x and y at their intersection point.
Step 2: Find the slope of the line perpendicular to the given line passing through the two given points.
- Use the formula: slope = (y2 - y1) / (x2 - x1)
Step 3: Use the point of intersection and the slope to write the equation in the general form: Ax + By + C = 0.
4) To calculate the vertex, x-intercept, y-intercept, and determine whether it represents a maximum or minimum, follow these steps:
Step 1: Identify the coefficients of the quadratic equation in the form y = ax^2 + bx + c.
- In the given equation, y = -3x^2 + 6x + 9, a = -3, b = 6, and c = 9.
Step 2: Calculate the x-coordinate of the vertex using the formula -b / (2a).
- Substitute the values of a and b into the formula to find the x-coordinate of the vertex.
Step 3: Substitute the x-coordinate of the vertex into the original equation to find the y-coordinate of the vertex.
Step 4: Use the quadratic formula to find the x-intercepts (zeros) of the equation.
- Set y = 0 in the quadratic equation and solve for x.
Step 5: Use the equation to find the y-intercept, which is the value of y when x = 0.
Step 6: Determine whether the vertex represents a maximum or minimum by looking at the coefficient a. If a is positive, the vertex represents a minimum and opens upwards. If a is negative, the vertex represents a maximum and opens downwards.
5a) The equation of a line parallel to the x-axis is of the form y = b, where b is the y-coordinate of the given point. In this case, b = -14. So, the equation of the line parallel to the x-axis passing through (8, -14) is y = -14.
b) To find the slope-intercept form equation between two points (-8, 14) and (-7, -3), follow these steps:
Step 1: Find the slope of the line using the formula: slope = (y2 - y1) / (x2 - x1).
Step 2: Substitute the slope and the coordinates of one of the points into the slope-intercept form equation y = mx + b. Solve for b to find the y-intercept.
Step 3: Write the final equation using the calculated slope and y-intercept.
c) To calculate the distance between two points (x1, y1) and (x2, y2), use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).
d) To calculate the midpoint between two points (x1, y1) and (x2, y2), use the midpoint formula:
midpoint = ((x1 + x2) / 2, (y1 + y2) / 2).