Problem 22 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$x-5=0$$

Step-by-Step Solution

Answer

The equation \(x - 5 = 0\) in standard form is \(x = 5\).

Step 1: Rewrite in Standard Form

In the given equation, \(x - 5 = 0\), we have just a single variable \(x\), so the standard form would be \(x = C\). All we need to do is rearrange the equation to express \(x\) as the subject of the equation.

Step 2: Rearrange the Equation

The equation can be rewritten as \(x = 5\).

Key Concepts

Integer Coefficients

In algebra, a coefficient is a constant by which a variable is multiplied. When we refer to 'integer coefficients', we're simply talking about these constants being whole numbers, which include the set of positive and negative numbers, as well as zero, but exclude fractions and decimals.

Taking the equation from the exercise, \(x - 5 = 0\), the coefficient in front of \(x\) is implied to be 1 (since \(x\) is the same as \(1x\)), and the constant term is -5. The challenge here is to rewrite the equation so that all coefficients are integers, which is already achieved in this case.

However, sometimes students might encounter equations with non-integer coefficients. For instance, \(0.5x = 2.5\) would not have integer coefficients. The goal, then, is to multiply by a common factor to clear the decimals, resulting in an equivalent equation with integer coefficients, like \(x = 5\). It’s essential in algebra to be comfortable with these conversions to solve problems in a standardized form.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. These expressions can involve numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. A well-known form is the linear equation, represented as \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

The exercise we have, \(x - 5 = 0\), is a simple algebraic equation which shows that when 5 units are added to a certain quantity \(x\), the result is 0. This form of equation is crucial for foundational algebra and appears frequently in various applications. Through such equations, students learn to manipulate expressions to find the value of unknown variables, a skill that is central to not just algebra, but all higher mathematics.

Understanding how to work with algebraic equations is essential for progressing through more complex mathematical concepts, including polynomials, inequalities, and even calculus.

Variable Isolation

Variable isolation refers to the process of rearranging an equation to solve for a single variable. This is one of the cardinal rules of algebra. The objective here is to have the variable on one side of the equation and the constant terms on the other. By doing so, we find the value of the variable that makes the equation true.

In the presented exercise, \(x - 5 = 0\), isolating the variable \(x\) is quite straightforward. All we need to do is add 5 to both sides of the equation, following the algebraic principle of maintaining equality by performing the same operation on both sides. The result, \(x = 5\), indicates that \(x\) is isolated, and we have found its value.

In more complex equations, the process might involve multiple steps, such as distributing, combining like terms, or using inverse operations to isolate the variable. These are the foundations upon which more advanced problem-solving skills are built, and mastering this concept is essential for success in algebra.