Algebra

a.2.b

To solve the equation (5 - 3(4x - 6) = 12), first distribute the -3:

[5 - 12x + 18 = 12.]

Then combine like terms:

[23 - 12x = 12.]

Subtract 23 from both sides:

[-12x = -11.]

Finally, divide by -12:

[x = \frac{11}{12}.]

A -1 slope in a graph indicates a line that descends from left to right at a 45-degree angle. For every unit you move one unit to the right along the x-axis, the line moves down one unit on the y-axis. This creates a straight line that consistently decreases, illustrating a negative relationship between the two variables plotted on the axes.

Three squared, written as (3^2), means multiplying 3 by itself. This results in (3 \times 3 = 9). Visually, if you were to represent it as a square, you would have a square with each side measuring 3 units, giving it an area of 9 square units.

The turning circle of a ship refers to the circular path the vessel takes when making a turn at a constant speed. The formula to calculate the turning circle radius (R) can be expressed as ( R = \frac{V^2}{g \cdot \tan(\theta)} ), where ( V ) is the ship's speed, ( g ) is the acceleration due to gravity, and ( \theta ) is the angle of heel or the angle of rudder deflection. The diameter of the turning circle is typically twice the radius.

Yes, ( y ) is typically considered a variable in mathematics and programming. It can represent a value that can change or vary within a given context, such as in equations, functions, or algorithms. Variables are essential for expressing relationships and performing calculations.

The expression demonstrates the distributive property, which states that a number multiplied by a sum can be distributed to each addend within the parentheses. In this case, 5 is multiplied by the sum of 3 and 5, resulting in 5 x 3 plus 5 x 5. This property allows us to simplify expressions and perform calculations more easily.

When solving a linear system by substitution, it's often best to choose the variable that is easiest to isolate. Look for a variable with a coefficient of 1 or -1, as this will simplify the process of rearranging the equation. If both equations are equally complex, consider which equation seems simpler to manipulate or offers fewer terms. Additionally, choose the variable that appears most frequently, as this can make the substitution process more efficient.

To solve the inequality ( x^2 < 9 ), we first rewrite it as ( x^2 - 9 < 0 ), which factors to ( (x - 3)(x + 3) < 0 ). The critical points are ( x = -3 ) and ( x = 3 ). Analyzing the intervals, we find that the solution to the inequality is ( -3 < x < 3 ). Therefore, the values of ( x ) that satisfy the inequality are those in the open interval ( (-3, 3) ).

The expression (2x^2 - 3x) is a quadratic polynomial in standard form, where (2x^2) represents the quadratic term and (-3x) is the linear term. To factor it, you can take out the common factor (x), resulting in (x(2x - 3)). This expression can be analyzed for its roots or further manipulated depending on the context.

Being able to solve a system of equations equips you with critical problem-solving and analytical skills, which are valuable in various fields such as engineering, economics, and finance. These skills help in making informed decisions by analyzing relationships between variables. Additionally, understanding systems of equations can improve logical reasoning and enhance your ability to tackle complex real-world problems. Overall, this mathematical foundation can be applicable in both personal and professional scenarios.

Equations originated from the need to describe relationships between quantities and to solve problems in mathematics and science. Early civilizations, such as the Babylonians and Egyptians, used simple arithmetic and geometric principles to create equations for trade, astronomy, and engineering. Over time, mathematicians like Diophantus, Al-Khwarizmi, and later figures in the Renaissance expanded these concepts, formalizing algebra and introducing symbolic notation. This evolution laid the foundation for modern mathematics and its applications across various fields.

The concept of cube roots has been known since ancient times, with various mathematicians contributing to its understanding. The Babylonians had methods for approximating cube roots as early as 2000 BCE. However, it is difficult to attribute the invention of the cube root to a single individual, as it evolved over centuries through the work of many cultures, including the Greeks and Indians. The formal notation and systematic study of roots developed much later, particularly during the Renaissance period.

To solve systems of equations by graphing, you plot each equation on the same coordinate plane and identify the point(s) where the lines intersect. The intersection point(s) represent the solution(s) to the system, indicating the values of the variables that satisfy both equations. If the lines intersect at one point, there is a unique solution; if they are parallel, there is no solution; and if they coincide, there are infinitely many solutions.

To complete the square for the expression (3x^2 - 12x + 9), first factor out the 3 from the first two terms: (3(x^2 - 4x) + 9). Next, to complete the square inside the parentheses, take half of (-4) (which is (-2)), square it (getting (4)), and then adjust the expression: (3(x^2 - 4x + 4 - 4) + 9). This simplifies to (3((x - 2)^2 - 4) + 9), resulting in (3(x - 2)^2 - 12 + 9), or finally (3(x - 2)^2 - 3).

To write a polynomial function with real coefficients given the zeros 2, -4, and (1 + 3i), we must also include the conjugate of the complex zero, which is (1 - 3i). The polynomial can be expressed as (f(x) = (x - 2)(x + 4)(x - (1 + 3i))(x - (1 - 3i))). Simplifying the complex roots, we have ((x - (1 + 3i))(x - (1 - 3i)) = (x - 1)^2 + 9). Thus, the polynomial in standard form is:

[ f(x) = (x - 2)(x + 4)((x - 1)^2 + 9). ]

Expanding this gives the polynomial (f(x) = (x - 2)(x + 4)(x^2 - 2x + 10)), which can be further simplified to the standard form.

The additive inverse of a number is the value that, when added to the original number, results in zero. For 0.75, the additive inverse is -0.75, since 0.75 + (-0.75) = 0.

The expression you provided seems to be a combination of terms rather than a complete equation. It appears to be missing an equals sign or additional context. If you meant to express a relationship or solve for a variable, please clarify or provide more details.

The square root of negative 128 can be expressed using imaginary numbers. First, we can rewrite it as (\sqrt{-128} = \sqrt{128} \cdot \sqrt{-1}). The square root of 128 simplifies to (8\sqrt{2}), so we have (\sqrt{-128} = 8\sqrt{2}i), where (i) is the imaginary unit. Thus, the simplified form is (8\sqrt{2}i).

Livicine is a chemical compound primarily derived from the plant species of the genus Livia. It is known for its potential anti-inflammatory and analgesic properties, which may contribute to pain relief and the reduction of inflammation in various medical applications. Additionally, livicine has been studied for its possible antioxidant effects, which can help protect cells from oxidative stress. However, further research is needed to fully understand its mechanisms and therapeutic potential.

The clavicle, or collarbone, serves several important functions in the human body. It acts as a strut that connects the arm to the body, providing stability and support to the shoulder joint. Additionally, it protects underlying structures such as nerves and blood vessels, and it serves as an attachment point for various muscles involved in shoulder and arm movement. Overall, the clavicle plays a crucial role in facilitating upper limb mobility and function.

An equation that has one solution is a linear equation of the form ( ax + b = c ), where ( a \neq 0 ). For example, the equation ( 2x + 3 = 7 ) has one solution: ( x = 2 ). This is because it can be rearranged to isolate ( x ) as a single value.

To find the slope of the line passing through the points (2.7, 1.4) and (2.4, 1.7), use the formula ( m = \frac{y_2 - y_1}{x_2 - x_1} ). Here, ( (x_1, y_1) = (2.7, 1.4) ) and ( (x_2, y_2) = (2.4, 1.7) ). Plugging in the values gives ( m = \frac{1.7 - 1.4}{2.4 - 2.7} = \frac{0.3}{-0.3} = -1 ). Therefore, the slope of the line is -1.

The mean ( x ) of a dataset can take any value within the range of the data, depending on the values of the individual data points. Specifically, it can be equal to the minimum value of the dataset if all data points are equal to this minimum, and it can approach the maximum value if the dataset includes very large values relative to other points. In a broader sense, the mean can be any real number, provided the dataset is appropriately constructed. It is also influenced by the number of data points and their distribution.

To find the value of X in the equation ( X^3 = 2744 ), you need to take the cube root of 2744. The cube root of 2744 is 14, since ( 14 \times 14 \times 14 = 2744 ). Therefore, ( X = 14 ).