Problem 94 - Beginning and Intermediate Algebra

Question

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \((5 x-2 y)(-2 a)\)

Step-by-Step Solution

Answer

-228

Step 1 - Substitute the Values

Substitute the given values of the variables into the expression Given: \( x = 6, y = -4, a = 3 \) Expression: \((5x - 2y)(-2a)\) Substitute: \((5 * 6 - 2 * (-4))(-2 * 3)\)

Step 2 - Simplify Inside the Parenthesis

Simplify the expression inside the parentheses \( (5 * 6) - (2 * (-4)) = 30 + 8 \) Since Multiplying a negative by a negative gives a positive

Hence, now we have \( (30 + 8)(-2*3)\)

Step 3 - Simplify Multiplication

Multiply and simplify the remaining values. \((30 + 8)(-2 * 3) → 38 * (-6)= -228\)

Key Concepts

Substitution

Let's start by substituting the given values into the expression. Substitution is the process of replacing each variable in an expression with its given value. In our problem, we have the expression \( (5x - 2y)(-2a) \). We need to evaluate this expression for \( x = 6, y = -4, \) and \( a = 3 \).

• First, we replace \( x \) with 6: \((5 * 6 - 2y)(-2a)\).
• Next, we replace \( y \) with -4: \((5 * 6 - 2 * (-4))(-2a)\).
• Finally, we replace \( a \) with 3: \((5 * 6 - 2 * (-4))(-2 * 3)\).

Now, we have the expression fully substituted: \((5 * 6 - 2 * (-4))(-2 * 3)\).

Simplification

After substitution, the next step is simplification. Simplification involves performing all the basic arithmetic operations, like addition, subtraction, multiplication, and division, in an orderly manner. Let's break it down:

• First, calculate inside the parentheses: \(5*6\) and \(-2*(-4)\).
• Perform the multiplications: \(5*6=30\) and \(-2*(-4)=8\).

When we multiply a negative number by another negative number, we get a positive number. So, \(-2 * (-4) = 8\).

Now our expression looks like this: \((30 + 8)(-2 * 3)\).

• Add inside the parenthesis: \(30 + 8 = 38\).

Now, the simplified expression is \((38)(-2*3)\).

Multiplication with Negative Numbers

The final step involves multiplication, particularly with a negative number. This calculation not only gives you the final result but also helps solidify your understanding of how negative numbers work. Let's take it step by step:

• Simplify the multiplication in the parentheses: \(-2 * 3 = -6\).
• Then, multiply the result with the number outside: \((38)(-6)\).
• Finally, perform the multiplication: \38 * (-6) = -228\.

Remember, when multiplying a positive number by a negative number, the result is always negative. Hence, \(38 * (-6) = -228\). This is our final answer!