define/illustrate fromula, rearrange and equation, and variables
Formula: A formula is a mathematical expression that represents a relationship between variables or quantities. It typically consists of symbols, numbers, and mathematical operators. Formulas are used to solve mathematical problems, make calculations, or describe scientific phenomena.
Example: The formula for the area of a rectangle is A = length × width, where A represents the area, and length and width are the dimensions of the rectangle. By plugging in the values for length and width, the formula allows us to find the area of any rectangle.
Rearrange an Equation: Rearranging an equation means rearranging its terms or variables to isolate a specific variable of interest. This is done by performing a series of mathematical operations in order to move terms from one side of the equation to the other.
Example: Consider the equation 2x + 5 = 15. To rearrange this equation to solve for x, we can subtract 5 from both sides: 2x + 5 - 5 = 15 - 5, which simplifies to 2x = 10. Then, dividing both sides by 2 gives x = 5. The equation has been rearranged to solve for x.
Variables: Variables are symbols or letters used to represent unknown quantities or changing values in mathematical equations or expressions. They can vary in value and are often used to denote relationships or patterns.
Example: In the equation y = mx + b, y and x are variables. The variable y represents the dependent variable, which depends on the value of the independent variable x. The variables m and b represent the slope and y-intercept, respectively, in the equation of a straight line. By substituting different values for x, we can find corresponding values for y and plot points to graph the equation.
Solve for each of the formulas for the indicated variable:
P = 2L + 2W for W
To solve the formula P = 2L + 2W for W, we need to isolate the variable W on one side of the equation. Here's how we can do it:
P = 2L + 2W
First, subtract 2L from both sides of the equation:
P - 2L = 2W
Next, we can divide both sides of the equation by 2 to solve for W:
(P - 2L) / 2 = W
So the formula for W in terms of P and L is:
W = (P - 2L) / 2
ax + b = 0 for x
To solve the equation ax + b = 0 for x, we will isolate the variable x on one side of the equation. Here's how we can do it:
ax + b = 0
First, subtract b from both sides of the equation:
ax + b - b = 0 - b
This simplifies to:
ax = -b
Next, divide both sides of the equation by a to solve for x:
(ax) / a = (-b) / a
This simplifies to:
x = (-b) / a
So the equation ax + b = 0, when solved for x, gives:
x = (-b) / a
V = LWH for H
To solve the formula V = LWH for H, we need to isolate the variable H on one side of the equation. Here's how we can do it:
V = LWH
First, divide both sides of the equation by LW:
V / (LW) = H
So the formula for H in terms of V, L, and W is:
H = V / (LW)