Problem 56 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value of the expression is 8.

Step 1: Substitute the Values into the Expression

First, substitute the given values of the variables into the expression. The given expression is \(3x - y + 4z\). Replace \(x\) with -2, \(y\) with 6, and \(z\) with 5. This gives us: \(3(-2) - 6 + 4(5)\).

Step 2: Perform the Multiplications

Next, calculate the multiplication operations. Compute \(3 \times (-2) = -6\) and \(4 \times 5 = 20\). Substitute these values back into the expression, which becomes \(-6 - 6 + 20\).

Step 3: Calculate the Addition and Subtraction

Finally, solve the expression by performing the operations from left to right. First, \(-6 - 6 = -12\). Then, add 20 to the result: \(-12 + 20 = 8\).

Key Concepts

Substitute Values into Expressions

To evaluate an algebraic expression, the first crucial step is to substitute the given values for the variables. This process involves replacing each variable in the expression with the specific number provided. Let's look at the expression provided: - Original expression: \(3x - y + 4z\)- Given values: \(x = -2\), \(y = 6\), \(z = 5\)Substitution results in replacing: - \(x\) with -2- \(y\) with 6- \(z\) with 5The expression now looks like this: \(3(-2) - 6 + 4(5)\). By systematically replacing the variables, you form a straightforward numeric expression that is easier to solve. Always double-check each substitution for accuracy, as any mistake could lead to incorrect results. Substituting values correctly sets the foundation for the solution.

Perform Multiplications in Expressions

Once you have substituted the values in the algebraic expression, the next step is to perform multiplications. This step simplifies the expression further by dealing with any multiplication operations present. For our expression, after substitution:- Compute \(3 \times (-2)\): This yields \(-6\).- Compute \(4 \times 5\): This results in \(20\).Now, substitute these results back into the expression: it changes to \(-6 - 6 + 20\). Clarifying the order of operations and performing each multiplication correctly is crucial. Multiplications can include integers (whole numbers) and should be calculated before moving forward. By following this order, you maintain the accuracy of the expression through its simplification.

Evaluate Expression: Calculate and Simplify

After multiplications are performed, you move on to calculate and simplify the expression by performing any remaining addition or subtraction. This is sometimes referred to as evaluating the expression.Starting from: \(-6 - 6 + 20\)- First, perform the subtraction: \(-6 - 6\) results in \(-12\).- Next, add \(20\): \(-12 + 20\) gives the final result of \(8\).It's important to approach this step with the correct order of operations, moving from left to right. Simplifying involves combining like terms and reducing the expression to its simplest form. By following these steps accurately—performing necessary calculations, and combining terms—you ensure the expression is fully simplified and accurately evaluated to its final value.