Problem 92 - Beginning and Intermediate Algebra

Question

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \(6 x-5 y+4 a\)

Step-by-Step Solution

Answer

The expression evaluates to 68.

Step 1: Substitute the Values

Substitute the given values of the variables into the expression: For \( x = 6, y = -4, a = 3 \), replace the variables in the expression \( 6x - 5y + 4a \).

Step 2: Expression After Substitution

After substituting the values, the expression becomes: \[ 6(6) - 5(-4) + 4(3) \].

Step 3: Simplify the Multiplications

Simplify the multiplications: \[ 36 + 20 + 12 \]

Step 4: Combine the Terms

Add the simplified terms together: \[ 36 + 20 + 12 = 68 \]

Key Concepts

Substitution in Algebra

Substitution in algebra involves replacing variables with their given numerical values. In this exercise, you are provided with specific values for the variables: x=6, y=-4, and a=3. By substituting these values into the original algebraic expression, \(6x - 5y + 4a\), you replace each variable with its corresponding number: \(6(6) - 5(-4) + 4(3)\). This step ensures you are working with numbers instead of variables, making the expression easier to evaluate.
Key points to remember:

• Always double-check that you have substituted the correct values.
• Pay close attention to the signs of the numbers provided (positive or negative).
• Write out each substitution step to avoid errors.

Simplification of Expressions

Simplification of expressions involves performing the operations within the algebraic expression to make it more manageable. After substituting the values into our expression, we have \(6(6) - 5(-4) + 4(3)\). To simplify this:

• First, multiply the numbers: \(6 \times 6 = 36\), \(5 \times -4 = -20\), and \(4 \times 3 = 12\).
• Note the signs when multiplying: a negative times a negative gives a positive.
• Record each simplified multiplier to keep track of progress: 36, 20 and 12.

Simplification reduces the expression into smaller, easier-to-handle parts. This prepares the expression for the final combination of terms.

Combining Like Terms

Combining like terms is the process of summing up the simplified parts of the expression. After performing the multiplication, we have the simplified terms: 36, 20, and 12. To complete our evaluation process, these terms need to be added together. Each term is a 'like term' as they are all constants. Add them step by step to avoid mistakes:

• First, add 36 and 20 to get 56.
• Then, add the result to 12: 56 + 12 = 68.

Combining like terms helps in reducing a complex expression into a single, easily understandable number. This final number represents the evaluated result of the original algebraic expression.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These fundamentals are crucial in evaluating algebraic expressions. In our exercise:

• We began with multiplication: \(6 \times 6\), \(5 \times -4\), and \(4 \times 3\). Correctly applying multiplication ensures each term's value is accurately calculated.
• Next, addition came into play. We combined the simplified multipliers: 36 + 20 + 12.
• Remember the importance of the sequence of operations. Always follow the order of operations (PEMDAS/BODMAS).
• Be mindful of negative signs, especially when they follow subtraction or multiplication rules.

Mastering basic arithmetic operations lays a strong foundation for tackling more complex algebraic problems efficiently and correctly.