Question

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

Short answer

k = 4

Step-by-step solution

Expand the Left Side

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

Set the Equation

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

Compare Coefficients

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

Solve for k

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

Key concepts

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.