Problem 18 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression when \(x=3, y=-2\), and \(z=4$$6 z+5 x-3 y\)

Step-by-Step Solution

Answer

The expression evaluates to 45 when \(x=3\), \(y=-2\), and \(z=4\).

Step 1: Substituting the given values

Substitute the values of \(x=3\), \(y=-2\), and \(z=4\) into the expression \(6z + 5x - 3y\). So, it becomes \(6*4 + 5*3 - 3*(-2)\).

Step 2: Simplification

Carry out the operations in the expression. So, it becomes \(24 + 15 + 6\).

Step 3: Calculation

Add the numbers from step 2 together to get the final solution. So, it becomes \(24 + 15 + 6 = 45\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra, used to evaluate expressions by replacing variables with their actual values. When you come across an algebraic expression like \(6z + 5x - 3y\), and you're given specific values for the variables, such as \(x=3\), \(y=-2\), and \(z=4\), the substitution method guides you step by step in the replacement process.

The first step is to take the expression and substitute each variable with the corresponding numerical value. In our specific case:
\[6(4) + 5(3) - 3(-2)\]
It's critical to replace the variables with care, especially when dealing with negative numbers. After substitution, always include the operation signs to avoid mistakes during the simplification process.

Simplifying Expressions

Once you've substituted the variables with numbers, the next step is simplifying the expression. Simplification is about performing arithmetic operations in the correct order following the rules of arithmetic, often referred to as the order of operations (PEMDAS/BODMAS).

In our example, after substitution, we get the numerical expression:\[6(4) + 5(3) - 3(-2)\]
This simplifies to:\[24 + 15 + 6\]
Here, no exponents or parentheses need further simplification, and there are no division or multiplication operations left to perform. So we proceed directly to addition and subtraction.

To avoid errors, it is crucial to work systematically, combining like terms and respecting arithmetic rules. Addition and subtraction are the final stages, which should be conducted left to right, yielding a simple numeric answer.

Algebraic Calculations

Algebraic calculations are the bread and butter of solving algebra problems, encompassing various arithmetic operations such as addition, subtraction, multiplication, and division. In the context of the given exercise, the algebraic calculations are fairly straightforward:

After simplifying the expression, we have:\[24 + 15 + 6\]
Adding these numbers together gives us the final result. Although this step may seem simple, it's essential not to rush and to double-check your calculations.

Summing up:\[24 + 15 = 39\]
\[39 + 6 = 45\]
So the final answer, after carrying out all algebraic calculations, is 45. Understanding every step and ensuring accuracy in calculations is vital as it builds a strong foundation for tackling more complex algebraic problems in the future.