Problem 24 - Algebra 1: Concepts and Skills

Question

What is the first step to prove the following theorem: If a and b are real numbers and \((\mathrm{x}+\mathrm{a})=\mathrm{b},\) then \(\mathrm{x}=\mathrm{b}-\mathrm{a}\). A. \(x+(a-a)=b-a\) B. \(x=b-a\) C. \((x+a)-a=b-a\) D. \(x+0=b-a\)

Step-by-Step Solution

Answer

The first step to prove the theorem is by taking away 'a' from both sides of the equation which will give (x + a) - a = b - a.

Step 1: Identify the initial problem

To initiate the process of proving, it is necessary to understand that the problem to prove is the theorem, which is: (x + a) = b implies x = b - a.

Step 2: Start the proof by isolating x

The process commences by taking the equation (x + a) = b and performing an arithmetic operation to isolate x on one side of the equation. By subtracting a from both sides of the equation, the left-hand side will be (x + a) - a and the right-hand side will be b - a.

Step 3: Simplify the equation

Now the equation is (x + a - a) = b - a. Simplify both sides to obtain x = b - a, because (a - a) equals to 0 and the sum of any element with 0 gives back the element.

Key Concepts

Algebraic Proofs

Algebraic proofs are logical sequences that establish the truth of a given mathematical statement. In algebra, proofs often involve showing that two expressions are equal under certain conditions. To approach an algebraic proof, one typically starts with a given equation or set of equations and applies a series of justified steps to arrive at the conclusion.

Each step must follow logically from the previous one and be based on established algebraic principles, such as the commutative, associative, and distributive properties. For the theorem in our exercise, the proof starts by showing \(\(x + a\) = b\) implies \(x = b - a\), a process which ultimately reinforces the student's understanding of how to manipulate and simplify algebraic expressions to isolate variables and simplify terms.

Isolation of Variables

Isolation of variables is a fundamental technique in algebra that involves rearranging an equation to solve for a specific variable. In the context of algebraic proofs, isolating a variable can demonstrate how different expressions are equivalent.

To isolate a variable, one typically performs operations such as addition, subtraction, multiplication, or division to both sides of an equation. This maintains the equation's balance while moving terms around so that the variable of interest is by itself on one side of the equation. For example, to prove the theorem from the exercise, isolating \(x\) is key. By subtracting \(a\) from both sides, the '+ a' and '- a' terms effectively cancel each other out, leaving \(x\) alone on one side.

Subtracting Like Terms

Subtracting like terms is a simplification process in algebra where terms with the same variable and exponent are combined by subtraction. This technique is used to reduce expressions to their simplest form.

In the provided exercise, subtracting like terms is the logical strategy used to simplify the left-hand side of the equation after isolating \(x\). Since \(a\) is added and then subtracted (which are like terms), they negate each other, effectively resulting in zero and leaving \(x\) by itself. Subtracting like terms effectively demonstrates that \(x + a - a\) simplifies to \(x\), which is a crucial step in completing the proof.