Algebra

To simplify the expression ((4x \cdot 7y - 5)(-7x - 2y + 1)), first distribute each term in the first expression to each term in the second expression. This results in multiple products: (4x \cdot (-7x)), (4x \cdot (-2y)), (4x \cdot 1), (-5 \cdot (-7x)), (-5 \cdot (-2y)), and (-5 \cdot 1). After performing the multiplication and combining like terms, the simplified expression is (-28x^2 - 8xy + 4x + 35x + 10y - 5), which can be further simplified to (-28x^2 + 27x + 10y - 5).

To calculate the number of license plates with 3 letters followed by 2 digits, we consider that there are 26 letters in the English alphabet and 10 digits (0-9). For the letters, there are (26^3) combinations, and for the digits, there are (10^2) combinations. Therefore, the total number of license plates is (26^3 \times 10^2 = 17,576,000).

A value that makes an equation "true" is known as a solution or root of the equation. When substituted into the equation, this value satisfies the equation, resulting in a true statement. For example, in the equation (x + 2 = 5), the value (x = 3) is a solution because substituting it yields a true statement: (3 + 2 = 5).

Let the two consecutive even integers be represented as ( n ) and ( n + 2 ). The difference between them is ( (n + 2) - n = 2 ). The absolute value of this difference is ( |2| ), which is simply 2. Therefore, the absolute value of the difference of two consecutive even integers is always 2.

To solve the equation (3x + 5 = 8), we first isolate (3x) by subtracting 5 from both sides, giving us (3x = 3). Dividing both sides by 3, we find (x = 1). Therefore, if we add 1 to (x), we get (x + 1 = 1 + 1 = 2).

No, when a binomial is squared, it results in a trinomial rather than a product with just two terms. Specifically, when you square a binomial of the form ( (a + b)^2 ), you expand it to ( a^2 + 2ab + b^2 ), which includes three distinct terms. Thus, the result of squaring a binomial cannot be expressed as a product with only two terms.

No, an exponential graph will never intersect the x-axis. This is because exponential functions, such as ( f(x) = a^x ) where ( a > 0 ), are always positive for all real values of ( x ). As ( x ) approaches negative infinity, the function approaches zero but never actually reaches it, meaning the graph will never touch or cross the x-axis.

"75 Y of M for the DAGA" likely refers to "75 years of membership for the Daughters of the American Revolution (DAR)." This signifies that an individual has been a member of the DAR for 75 years, celebrating a significant milestone in their commitment to the organization's mission of promoting historic preservation, education, and patriotism.

When solving a radical equation, the first step is to isolate the radical expression on one side of the equation. After isolating the radical, you can eliminate it by squaring both sides of the equation. This process may introduce extraneous solutions, so it's important to check all potential solutions in the original equation. Always ensure the final answers satisfy the original radical equation.

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.