Problem 3 - Algebra Form and Function

Question

Is the expression linear? $$ 6 r+r-1 $$

Step-by-Step Solution

Answer

Answer: Yes, the expression is linear.

Step 1: Combine like terms

Combine the terms with the variable \(r\) to simplify the expression: $$ 6r + r - 1 $$ can be simplified as $$ (6 + 1)r - 1 $$

Step 2: Final expression

After simplifying, the given expression becomes: $$ 7r - 1 $$

Step 3: Check for linearity

Now, let's check if the expression is linear. Here are the two criteria to be fulfilled: 1. The variable (\(r\)) has a power of 1. 2. The expression has a constant term (in this case, \(-1\)). Both conditions are met in our expression, \(7r - 1\).

Step 4: Conclusion

Since both linearity criteria are fulfilled, we can conclude that the given expression, \(6r + r - 1\), is indeed linear.

Key Concepts

Combining Like Terms

When working with linear expressions, one of the key skills is to combine like terms. Like terms are terms that contain the same variables, raised to the same power. For example, in the expression \(6r + r - 1\), the terms \(6r\) and \(r\) are like terms because they both contain the variable \(r\) raised to the power of 1.
Combining like terms is a simplifying process where you add or subtract coefficients of similar terms. By combining \(6r\) and \(r\), you effectively add their coefficients:

• The coefficient of \(6r\) is 6.
• The coefficient of \(r\) is 1 (since \(r\) is the same as \(1r\)).

Adding these coefficients together gives you \((6+1)r = 7r\). Therefore, the expression \(6r + r\) simplifies to \(7r\). Learning how to combine like terms helps simplify expressions and solve equations more efficiently.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves rewriting expressions in a more compact, simpler form while maintaining their original value. For example, after combining like terms in the expression \(6r + r - 1\), you get a simplified version: \(7r - 1\).
The main goal of simplifying is to make expressions easier to interpret and solve. Simplification is usually achieved through a series of operations, including combining like terms, applying the distributive property, and reducing fractions when applicable. Each method has specific steps and purposes. However, they all aim to reduce complex expressions to their simplest form.By practicing simplification, you can more easily recognize patterns and relationships within algebraic expressions, which is especially helpful when dealing with more complex algebra problems.

Linearity Criteria

To determine if an expression is linear, you must check if it meets the linearity criteria. Linear expressions typically have variables raised to the power of one and may include a constant term. These are essential components that tell you if an expression forms a straight line when graphed.
For our given expression \(7r - 1\), you need to consider:

• The power of the variable \(r\). In this case, \(r\) is raised to the power of 1, which meets the first criterion.
• The presence of a constant term. Here, the constant term is \(-1\), satisfying the second condition.

Since both criteria are met, the expression \(7r - 1\) is linear. Recognizing these criteria is crucial in algebra because linear equations have unique properties and are often easier to work with than nonlinear ones. Identifying linearity helps determine how to solve and interpret problems effectively.