Algebra

To simplify the expression (3(7x + 2y)), you distribute the 3 to both terms inside the parentheses. This results in (3 \times 7x + 3 \times 2y), which simplifies to (21x + 6y). Thus, the final answer is (21x + 6y).

The inverse of the function ( F(x) = bx ) is ( F^{-1}(x) = \frac{x}{b} ), provided that ( b \neq 0 ). This is derived by solving the equation ( y = bx ) for ( x ), which gives ( x = \frac{y}{b} ). Thus, the inverse function effectively "undoes" the multiplication by ( b ).

The cuspid, also known as a canine tooth, plays a crucial role in the human mouth by aiding in the tearing and piercing of food during chewing. Positioned between the incisors and premolars, cuspids have a sharp point that helps grasp and hold onto food, making them essential for effective digestion. Additionally, they contribute to the alignment of the dental arch and help maintain the structure of the mouth and face.

0.3 is bigger than 0.03. This is because 0.3 can also be expressed as 3/10, while 0.03 is equivalent to 3/100. Since 3/10 is greater than 3/100, it follows that 0.3 is the larger number.

The abscissa is commonly referred to as the "x-coordinate" in a Cartesian coordinate system. It represents the horizontal distance from the origin to a point on a graph. Additionally, in some contexts, it may simply be called the "horizontal coordinate."

The verbal expression for ( h^3 ) is "h cubed" or "h raised to the third power." It represents the multiplication of the variable ( h ) by itself three times: ( h \times h \times h ).

A ball joint is a pivotal component in a vehicle's suspension system, allowing for smooth movement and flexibility between the wheel and the chassis. It enables the wheel to pivot while providing support, accommodating vertical and horizontal motion as the vehicle navigates bumps and turns. This articulation helps maintain proper wheel alignment and contributes to overall vehicle stability and handling. Additionally, ball joints are crucial for ensuring a comfortable ride by absorbing shocks and vibrations from the road.

If a 4-digit personal identification number cannot contain a zero, each digit can be any of the numbers from 1 to 9, giving us 9 possible choices for each digit. Therefore, the total number of possible 4-digit PINs is calculated as (9 \times 9 \times 9 \times 9), which equals (9^4). This results in a total of 6,561 different possible 4-digit PINs.

The function ( g(x) = 3(x-2)^4 ) has undergone two transformations from its parent function ( f(x) = x^4 ). First, it has been horizontally shifted to the right by 2 units due to the ( (x-2) ) term. Second, it has been vertically stretched by a factor of 3 because of the coefficient 3 in front of the function.

A point lies on the x-axis if its y-coordinate is zero, meaning it can be represented as ((x, 0)). Conversely, a point lies on the y-axis if its x-coordinate is zero, represented as ((0, y)). If either condition is met, the point is on the respective axis.

Yes, (5 - 7d) is not a monomial; it is a binomial because it consists of two terms: (5) and (-7d). A monomial is defined as an algebraic expression that contains only one term, which can include constants, variables, or their products. In contrast, a binomial has two distinct terms separated by a plus or minus sign.

A graph of a function cannot have two different coordinates (or points) with the same x-value because, by definition, a function assigns exactly one output (y-value) for each input (x-value). If a graph did have two points with the same x-coordinate but different y-coordinates, it would violate the definition of a function, as a single input would yield multiple outputs. This concept is often referred to as the "vertical line test," where any vertical line drawn on the graph intersects it at most once.

To solve the equation (5x - x + 12 + 2x - 710 = 0), first simplify the left side: combine like terms. This gives (6x - 698 = 0). Adding 698 to both sides results in (6x = 698), and dividing by 6 yields (x = \frac{698}{6} = 116.33). Thus, the solution is (x \approx 116.33).

The square root of the expression (2x + 1) is written as (\sqrt{2x + 1}). If you meant to include the number 3 in the expression, please clarify how it relates to (2x + 1). Otherwise, (\sqrt{2x + 1}) represents the principal square root of the expression.

To graph the equation (4X + 5Y = 20), first rearrange it into slope-intercept form (Y = mx + b). This can be done by isolating (Y): (5Y = -4X + 20) or (Y = -\frac{4}{5}X + 4). Next, plot the y-intercept (0, 4) on the graph, and use the slope of (-\frac{4}{5}) to find another point by going down 4 units and right 5 units. Finally, draw a straight line through these points to represent the equation.

The equation ( V = IR ) is known as Ohm's Law. It represents the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit. According to Ohm's Law, the voltage across a conductor is directly proportional to the current flowing through it, with resistance as the proportionality constant. This fundamental principle is widely used in electrical engineering and physics.

The order of operations, a set of rules for determining the sequence in which mathematical operations should be performed, was formalized by mathematicians over time rather than being introduced by a single individual. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is often used to help remember the order. Although the exact origins are not attributed to a specific person, the conventions became standardized in the 19th century as mathematics evolved.

No, the slope of a line is not the coefficient of the y-term in the line's equation. In the slope-intercept form of a line's equation, (y = mx + b), the slope is represented by the coefficient (m) of the x-term, while (b) represents the y-intercept. Therefore, the slope corresponds to the x-term's coefficient, not the y-term.

The square root of 112 is approximately 10.583. To the nearest whole number, this value rounds to 11.

Problem-solving should be used in teaching when educators aim to enhance critical thinking and analytical skills among students. It is particularly effective in subjects that require applying concepts to real-world situations, fostering deeper understanding and retention. Additionally, employing problem-solving strategies can engage students actively, encouraging collaboration and communication as they work through challenges together. Overall, integrating problem-solving into lessons supports a more interactive and meaningful learning experience.

The vertex of a parabola is the highest point if the parabola opens downward, making it a maximum point. Conversely, if the parabola opens upward, the vertex is the lowest point, known as a minimum. Thus, whether the vertex is the highest or lowest point depends on the direction in which the parabola opens.

To find (210^2 + 150^2), first calculate each square: (210^2 = 44100) and (150^2 = 22500). Adding these together gives (44100 + 22500 = 66600). Therefore, (210^2 + 150^2 = 66600).

To find the number that squared equals 2304, you need to take the square root of 2304. The square root of 2304 is 48, since 48 multiplied by itself (48 × 48) equals 2304. Therefore, 48 squared equals 2304.

Amontons' equation describes the relationship between pressure and temperature in gases, formulated by French physicist Guillaume Amontons in the 17th century. It states that for a given mass of gas at constant volume, the pressure is directly proportional to its absolute temperature. Mathematically, it can be expressed as ( P \propto T ), where ( P ) is the pressure and ( T ) is the absolute temperature. This principle is foundational in thermodynamics and helps explain gas behavior under various conditions.

Linear equations can be traced back to ancient civilizations, with some of the earliest known uses found in Babylonian mathematics around 2000 BCE. The Babylonians developed methods to solve linear equations, often in the context of problems related to trade and land measurement. Additionally, ancient Greek mathematicians like Euclid further explored these concepts. However, the formalization of linear equations as we understand them today evolved over centuries, particularly during the Renaissance and the development of algebra.