Problem 86 - Algebra and Trigonometry

Question

Evaluate \(x^{2}-x\) for the value of \(x\) satisfying $$2(x-6)=3 x+2(2 x-1)$$

Step-by-Step Solution

Answer

The value of the expression \(x^{2}-x\) for the given value of \(x\) is 6.

Step 1: Simplify both sides of the equation

First, simplify both sides of the equation \(2(x-6)=3x+2(2x-1)\). Distribute the factors on both sides to get \(2x - 12= 3x + 4x - 2\).

Step 2: Collect like terms

Second, collect like terms on both sides of the equation. On the right-hand side, \(3x + 4x\) gives \(7x\), giving us \(2x - 12= 7x - 2\).

Step 3: Solve for x

Isolate the x term on the one side and the constant term on the other side. This leads us to \(2x - 7x = -2 + 12\), or \(-5x = 10\). Solving for \(x\) from this equation gives us a value of \(x = -2\).

Step 4: Substitute into the expression

Finally, substitute the obtained value of \(x\) into the expression \(x^{2}-x\). On substituting \(x = -2\) into the expression, we get \((-2)^2 - (-2) = 4 + 2 = 6\)

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. For example, in the expression \(x^2 - x\), \(x\) represents a variable, which can take on different values, while the operations include squaring (raising to the power of two) and subtraction.

Understanding how to work with these expressions is essential for solving equations, as they represent the relationship between the variable and the constants. When evaluating an algebraic expression, like \(x^2 - x\), one must substitute the variable \(x\) with a specific number and perform the arithmetic operations as indicated.

Collect Like Terms

To collect like terms means to combine terms in an algebraic expression that have the same variables raised to the same powers. This simplifies the expression, making it easier to solve or manipulate. For instance, if you have \(3x + 4x\), you can combine these to get \(7x\) because both terms have the variable \(x\) to the same power.

Collecting like terms is a critical step in solving linear equations. It allows us to rewrite equations in a simpler form, as seen in the textbook problem where the equation \(2x - 12 = 3x + 4x - 2\) simplifies to \(2x - 12 = 7x - 2\) after collecting like terms. This simplification then facilitates the subsequent processes of isolating the variable and solving the equation.

Substitution Method

The substitution method is a technique used to find the value of a variable in an equation. It involves solving one equation for one variable, then substituting this solution back into another expression or equation. This method can be used to solve systems of equations or to evaluate expressions at a particular value.

For example, once the value of \(x\) is determined to be \(x = -2\) using the previous steps in the textbook solution, the substitution method is used to find the value of the algebraic expression \(x^2 - x\) by replacing \(x\) with -2. The resulting calculation, \( (-2)^2 - (-2) \), demonstrates how the expression simplifies to 6 when evaluated at the given value of \(x\). This substitution gives a tangible result for the originally abstract expression.