Problem 46 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(-4 x+9 y-3 x-y, \quad x=-4\) and \(y=7\)

Step-by-Step Solution

Answer

The expression evaluates to 84.

Step 1: Substitute the variables

First, substitute the given values of the variables into the expression. The expression is \(-4x + 9y - 3x - y\). Substitute \(x = -4\) and \(y = 7\) into the expression: \[-4(-4) + 9(7) - 3(-4) - 7\].

Step 2: Calculate each term

Now, let's calculate each term of the expression separately to simplify the process: 1. \(-4(-4) = 16\) 2. \(9(7) = 63\)3. \(-3(-4) = 12\)4. \(-y = -7\)

Step 3: Combine like terms

With each term calculated, combine them together: \[16 + 63 + 12 - 7\].

Step 4: Simplify the result

Add and subtract the terms from Step 3: 1. \(16 + 63 = 79\)2. \(79 + 12 = 91\)3. \(91 - 7 = 84\). Therefore, the value of the expression is \(84\).

Key Concepts

Substitution Method

The substitution method is often the first step in evaluating algebraic expressions. This technique involves replacing variables in an expression with specific values given in the problem.
To substitute effectively:

• Identify each variable in the expression.
• Replace each variable with the corresponding value provided.
• Use parentheses to ensure proper handling of negative signs and multiplication.

For instance, in the given problem, we substitute \(x = -4\) and \(y = 7\) into the expression \(-4x + 9y - 3x - y\). After substitution, it becomes \(-4(-4) + 9(7) - 3(-4) - 7\). This step sets up the expression for further evaluation by replacing abstract variables with concrete numbers.

Combining Like Terms

Once you've substituted the variables, the next step is to combine like terms. This means adding or subtracting terms that have identical variable components or parts.
When combining like terms:

• Look for terms with the same variable. For example, in \(-4x - 3x\), "x" is the shared variable.
• Add or subtract the coefficients of these terms.
• Remember that terms without variables, like constants, can be combined separately.

In our problem, after substitution, we focus on the numbers obtained: \(16\) from \(-4(-4)\), \(63\) from \(9(7)\), \(12\) from \(-3(-4)\), and \(-7\) from \(-y\). These are combined in the next step to simplify the expression.

Simplifying Expressions

The final step in evaluating an expression is simplifying it by performing the arithmetic operations. This involves:

• Calculating the sum or difference of all combined terms.
• Ensuring calculations are done correctly to prevent errors.

For instance, in our example, we combine \(16 + 63 + 12 - 7\). We start by adding the positive numbers: \(16 + 63 = 79\) and then \(79 + 12 = 91\). Finally, we subtract \(-7\), giving us the final result of \(91 - 7 = 84\).
By carefully simplifying the expression in steps, you'll reach the correct mathematical evaluation. This process helps build confidence and understanding in dealing with algebraic expressions.