Problem 32 - Elementary Algebra

Question

Solve. $$ -7 b+3=2-5 b+1-2 b $$

Step-by-Step Solution

Answer

The equation simplifies to a true statement, so \(b\) can be any real number.

Step 1: Simplify the Equation

First, simplify both sides of the equation by combining like terms. On the right-hand side, combine \(-5b\) and \(-2b\), which gives \(-7b\). Also, combine constants 2 and 1 to get 3. Thus, the equation becomes:\[-7b + 3 = 3 - 7b\]

Step 2: Remove Like Terms

Notice that \(-7b\) appears on both sides of the equation. Add \(7b\) to both sides of the equation to eliminate \(b\) terms from the equation:\[-7b + 7b + 3 = 3 - 7b + 7b\]Simplifying both sides results in:\[3 = 3\]

Step 3: Analyze the Equation's Result

The equation simplifies to a true statement \(3 = 3\). This means that the original equation holds true for any real number \(b\). Thus, \(b\) can be any real number.

Key Concepts

Solving Equations

Solving algebraic equations involves finding values for variables that make the equation true. When faced with an equation, start by simplifying both sides as much as possible.

• Combine like terms: Variables and constants should be grouped together separately for easier management.
• Perform operations: Add, subtract, multiply, or divide both sides by the same number as needed to isolate the variable.

After simplification, consider the new form of the equation. Sometimes, you can directly solve for the variable, while other times, more simplifying may be necessary.

This exercise, for instance, resulted in a simplified expression that was actually a statement of equality. This indicates the original equation is an identity, meaning it is true for any value of the variable. Identifying these situations helps in recognizing cases where no specific solution exists, but rather a set of infinite solutions.

Like Terms

Understanding like terms is essential when simplifying equations. Like terms have the same variable factor with the same exponent. For example, \(-5b\) and \(-2b\) are like terms because they both contain the variable \(b\).

• Combine coefficients: The coefficients (numerical part of terms) of like terms can be summed or subtracted while the variable factor remains unchanged.
• Reduces complexity: By reducing the number of terms, equations become much simpler to solve.

In the exercise, combining \(-5b\) and \(-2b\) reduced these to \(-7b\), simplifying handling on the right-hand side. Remember that even constant terms can be like terms with each other, as they have no variable parts to differentiate them.

Real Numbers

In algebra, real numbers are the set of all rational and irrational numbers, encompassing all the possible cases for solutions to equations. They include:

• Integers: Whole numbers and their negatives, such as -3, 0, 4.
• Fractions and decimals: Any part of a whole, like 1/3 or 2.75.
• Irrational numbers: Numbers that cannot be expressed as exact fractions, such as \(\sqrt{2}\) or \(\pi\).

Understanding that every real number could potentially be a solution is vital. For the equation in the exercise, it simplified to \(3 = 3\), indicating an identity. Thus, the solution encompasses every real number since there were no restrictions placed by the original equation's structure. Recognizing these situations helps acknowledge the broad applicability of solutions to variable equations.