Algebra

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A line that is parallel to the line represented by the equation ( y = -3x - 7 ) will have the same slope. Since the slope of the given line is -3, any parallel line will also have a slope of -3. Therefore, an equation representing a parallel line can be written in the form ( y = -3x + b ), where ( b ) is any real number.

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To calculate (5i^2), we start by recalling that (i) is the imaginary unit, defined as (i = \sqrt{-1}). Therefore, (i^2 = -1). Thus, (5i^2 = 5 \times (-1) = -5).

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A square root can be represented by the rational exponent of ( \frac{1}{2} ). For any non-negative number ( x ), the square root is expressed as ( x^{1/2} ). This means that taking the square root of ( x ) is equivalent to raising ( x ) to the power of ( \frac{1}{2} ).

To solve a proportion, you can use the cross-multiplication method. If you have a proportion in the form ( \frac{a}{b} = \frac{c}{d} ), you can set up the equation ( a \times d = b \times c ). This allows you to find the unknown variable in the proportion by rearranging the equation as needed.

To find the x-intercepts of the parabola described by the equation (x^2 + 4x - 12 = 0), we can use the quadratic formula (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}). Here, (a = 1), (b = 4), and (c = -12). Calculating the discriminant (b^2 - 4ac) gives (16 + 48 = 64). Therefore, the x-intercepts are (x = \frac{-4 \pm 8}{2}), resulting in (x = 2) and (x = -6).

The algebraic expression for 7 decreased by 4 times a number can be written as ( 7 - 4x ), where ( x ) represents the number. This expression captures the operation of subtracting four times the variable from seven.

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It seems there is a small error in the equation you provided. If the equation is meant to be ( y = x - 10.9 ), then the y-intercept is the value of ( y ) when ( x = 0 ). Plugging in ( x = 0 ) gives ( y = 0 - 10.9 = -10.9 ). Thus, the y-intercept is (-10.9). If the equation is different, please clarify for accurate assistance.

The smallest power of 10 that exceeds 987654321098765432 is (10^{18}). This is because (10^{18} = 1,000,000,000,000,000,000), which is greater than 987654321098765432, while (10^{17} = 100,000,000,000,000,00) is less than it. Therefore, (10^{18}) is the answer.

A polynomial of degree 5 can have up to 5 zeros, counting multiplicities. This means it can have fewer than 5 distinct zeros if some of them are repeated. According to the Fundamental Theorem of Algebra, a polynomial of degree ( n ) has exactly ( n ) roots in the complex number system, including real and non-real roots.

To simplify the expression ( 6(-2.3x - 5) + (4x + 11) ), first distribute the 6:

( 6(-2.3x) = -13.8x ) and ( 6(-5) = -30 ), giving us ( -13.8x - 30 ).

Now add ( 4x + 11 ):

Combine like terms: ( -13.8x + 4x = -9.8x ) and ( -30 + 11 = -19 ).

Thus, the final expression is ( -9.8x - 19 ).

The slope of a line, denoted by ( m ), indicates how steep the line is and the direction it goes. If ( m = -4 ), this means the line descends at a steep angle, decreasing 4 units vertically for every 1 unit it moves horizontally to the right. A negative slope indicates that as the ( x )-value increases, the ( y )-value decreases. Thus, the line will trend downward from left to right.

To find a quadratic function that models the relationship between the independent and dependent variables, you can use methods such as polynomial regression if you have data points, or you can utilize the standard form (y = ax^2 + bx + c) to determine the coefficients (a), (b), and (c). This can be achieved by fitting the data to the quadratic form using techniques like least squares or by using vertex and intercept forms if specific points or features of the graph are known. Additionally, you can also use systems of equations if you have specific points through which the parabola passes.

The difference of cubes formula is an algebraic identity that expresses the difference between the cubes of two terms. It is given by the formula: ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) ). This formula allows you to factor the difference of cubes into a linear factor and a quadratic factor. It is useful for simplifying expressions and solving equations involving cubic terms.

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The standard form of a quadratic equation is expressed as ( ax^2 + bx + c = 0 ), where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ). The general form is similar but often written as ( f(x) = ax^2 + bx + c ) to represent a quadratic function. Both forms highlight the parabolic nature of quadratic equations, with the standard form emphasizing the equation set to zero.

Non-examples of linear equations include any equations that involve variables raised to powers other than one, such as quadratic equations (e.g., (y = x^2 + 3)) or cubic equations (e.g., (y = x^3 - 2x)). Additionally, equations that include products of variables (e.g., (y = xy + 1)) or functions like sine or cosine (e.g., (y = \sin(x))) are also non-linear. Graphically, these equations will not produce straight lines but rather curves or more complex shapes.

Problem representation is crucial in the problem-solving process because it defines how a problem is understood and framed, influencing the strategies employed to tackle it. A clear and accurate representation can simplify complex issues, highlight key components, and guide the solver towards relevant solutions. If a problem is misrepresented, it can lead to ineffective or misguided approaches, ultimately hindering successful outcomes. Thus, effective problem representation sets the foundation for successful problem-solving.

First-order differential equations have numerous applications across various fields. In physics, they can describe processes such as radioactive decay and population dynamics, where the rate of change of a quantity is proportional to its current value. In engineering, they are used to model systems like electrical circuits and fluid flow. Additionally, in economics, they can help analyze growth models and investment strategies, capturing how variables evolve over time.

The square root of 74 is irrational. This is because 74 is not a perfect square, and its square root cannot be expressed as a fraction of two integers. Thus, it does not meet the criteria for being a rational number.

The equation you've provided, 5 - 3x45, appears to be a mix of numbers and a variable. If you meant to represent it as (5 - 3x \cdot 45), then you need to simplify it. In that case, the coefficient of (x) would be (-3 \cdot 45), which equals (-135). If you meant something else, please clarify!