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Chapter 0: Problem 101

Find \(k\) if \(3 x(x-5 k)=3 x^{2}-60 x\).

Short Answer

Expert verified
k = 4

Step by step solution

01

Expand the Left Side

Expand the left side of the equation: \(3x(x - 5k)\) Distribute 3x in the parentheses which gives: \[3x^2 - 15kx\]
02

Set the Equation

Now, the equation looks like: \[3x^2 - 15kx = 3x^2 - 60x\]
03

Compare Coefficients

The terms \(3x^2\) on both sides are equal. Now, compare the coefficients of \(x\): \[-15k = -60\]
04

Solve for k

Solve for \(k\) by dividing both sides by -15: \[k = \frac{-60}{-15} = 4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
An algebraic equation is a mathematical statement where two expressions are set equal to each other. These expressions can include variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. For example, our exercise involves the equation: \[ 3x(x - 5k) = 3x^{2} - 60x \] The goal is to find the unknown variable, which in this case is \(k\). Breaking down and simplifying these equations is a fundamental skill in algebra.
Coefficient Comparison
Coefficients are the numerical factors that multiply the variables in an equation. For instance, in the term \( 3x^2 \), the number 3 is the coefficient of \( x^2 \). Comparing coefficients involves looking at the terms on both sides of an equation and matching up the coefficients of like terms (terms with the same variables and powers). In our exercise, after expanding both sides, we got: \[ 3x^2 - 15kx = 3x^2 - 60x \] Here, we compare the coefficients of \( x \) , leading to the equation: \[ -15k = -60 \]
Distributive Property
The distributive property is a fundamental rule in algebra that allows you to multiply a single term by each term inside a parenthesis. It is expressed as: \[ a(b + c) = ab + ac \] In our exercise, we applied the distributive property to \( 3x(x - 5k) \). This means we'll distribute \( 3x \) across \( x \) and \( -5k \), giving us: \[ 3x(x - 5k) = 3x \times x + 3x \times (-5k) = 3x^2 - 15kx \] This step is crucial for simplifying and eventually solving the algebraic equation.
Solving for a Variable
To solve for a variable means to isolate that variable on one side of the equation. In our problem, once we had \[ -15k = -60 \] we needed to isolate \(k\). We did this by performing the same operation on both sides of the equation. Dividing both sides by \(-15\), we get: \[ k = \frac{-60}{-15} = 4 \] Thus, the value of \(k\) is 4. This process of isolating the variable is a key skill in algebra and foundational for solving various mathematical problems.

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Most popular questions from this chapter

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{\sqrt{1+x}-x \cdot \frac{1}{2 \sqrt{1+x}}}{1+x} \quad x>-1$$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(16 x^{2} y^{-1 / 3}\right)^{3 / 4}}{\left(x y^{2}\right)^{1 / 4}} $$

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{5-\sqrt{43}}{3}$$

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$2 x\left(x^{2}+1\right)^{1 / 2}+x^{2} \cdot \frac{1}{2}\left(x^{2}+1\right)^{-1 / 2} \cdot 2 x$$

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x-7}-1}{x-8} \quad x \neq 8$$
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