maths
Explain how to check if a given point is a solution to a linear equation.
Substitute the coordinates of the point into the equation and verify if the equation holds true.
If the number of hours worked (h) is directly proportional to earnings (E), and someone earns $120 for 8 hours, how much would they earn for 10 hours?
$150
What type of function is represented by a linear equation?
Linear function
In the equation y = kx, what does 'k' represent?
The constant of proportionality
What is the y-intercept of the linear equation 4x - 2y = 8?
-4
Determine the midpoint of the segment connecting points (1, 5) and (7, 9).
(4, 7)
Describe the key characteristic of a linear equation's graph.
A straight line
Solve the linear equation: 3x - 7 = 11
x = 4
If a line segment has endpoints A and B, and M is its midpoint, what can you say about the lengths of AM and MB?
They are equal.
Provide the formula for the x-coordinate of the midpoint between two points (x1, y1) and (x2, y2).
(x1 + x2) / 2
Write the mathematical expression for 'y is directly proportional to x'.
y = kx
How is the slope of a horizontal linear equation different from that of a vertical linear equation?
A horizontal line has a slope of 0, while a vertical line has an undefined slope.
How can you tell from a graph if two quantities are in direct proportion?
The graph is a straight line passing through the origin (0,0).
Calculate the midpoint of the line segment with endpoints (-3, 2) and (5, -4).
(1, -1)
What is the general form of a linear equation in two variables?
Ax + By = C
Find the value of x that satisfies the equation 5(x + 2) = 3x + 16.
x = 3
What does the midpoint represent geometrically?
The exact middle point of a line segment.
Describe a real-world situation where finding a midpoint might be useful.
Locating a central meeting point between two cities on a map, or finding the center of a symmetrical object.
If the midpoint of a segment is (2, 3) and one endpoint is (0, 1), what are the coordinates of the other endpoint?
(4, 5)
Find the x-intercept of the equation 2x + 3y = 12.
x = 6
If M is the midpoint of segment AB, and A=(x1, y1), B=(x2, y2), what is the relationship between vector AM and vector MB?
They are equal (AM = MB).
What happens to y if x is doubled in a direct proportion relationship?
y is also doubled.
What is another name for a direct proportion relationship?
Direct variation
Given points P(a, b) and Q(c, d), what is the formula for the y-coordinate of their midpoint?
(b + d) / 2
If two variables are in direct proportion, what can you say about their ratio?
Their ratio is constant.
Explain the difference between direct proportion and inverse proportion.
In direct proportion, two quantities increase or decrease together at a constant ratio (y=kx). In inverse proportion, as one quantity increases, the other decreases, and their product is constant (y=k/x).
What is the slope of the line represented by the equation y = -2x + 5?
-2
If the cost of apples (C) is directly proportional to their weight (W), and 2 kg cost $4, how much would 5 kg cost?
$10
Which of the following equations represents a direct proportion?
y = 3x
If y is directly proportional to x, and y=15 when x=3, find the constant of proportionality.
k = 5