Problem 35 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(3 x+7 y, \quad x=-1\) and \(y=-2\)

Step-by-Step Solution

Answer

The evaluated expression is -17.

Step 1: Substitute the Values into the Expression

The given algebraic expression is \(3x + 7y\). Substitute \(x = -1\) and \(y = -2\) into the expression: \[ 3(-1) + 7(-2) \].

Step 2: Multiply the Constants and Variables

Calculate the multiplication for each term: \(3(-1) = -3\) and \(7(-2) = -14\).

Step 3: Add the Results

Add the results from the multiplications: \(-3 + (-14) = -17\).

Key Concepts

Variable Substitution

In algebra, substituting variables with their specific values is a fundamental skill. For the expression given, which is \(3x + 7y\), we need to substitute \(x = -1\) and \(y = -2\). This means wherever you see \(x\) in the expression, you replace it with \(-1\), and wherever you see \(y\), you substitute it with \(-2\). So, the expression \(3x + 7y\) becomes \(3(-1) + 7(-2)\).

• This step is crucial because it transforms the expression from an abstract representation into specific numeric terms you can calculate.
• A key point is to ensure you substitute the correct values for each variable consistently throughout the expression.

Step-by-Step Solutions

When tackling any algebraic expression, working through each component in steps is beneficial. In our example, after substituting the variables, each term is evaluated separately. This step-by-step approach avoids errors and helps in understanding each operation clearly.

• Begin by substituting the values into the expression.
• Next, perform the calculations on each term individually.
• Lastly, combine the simplified results to get the final answer.

This methodical breakdown makes complex problems much more manageable, building confidence with mathematics one step at a time.
It is like solving little puzzles that come together to form a complete picture.

Multiplication of Constants

Once the variable substitution is complete, you proceed to multiply constants by the substituted variable values. Algebraic expressions often involve multiplying coefficients (constants) with variable values.

• In our example, these actions occur in each term: \(3\times(-1)\) and \(7\times(-2)\).
• Careful note should be taken of signs. A negative times a positive results in a negative, and a negative times a negative results in a positive. Multiply following these rules to avoid errors.

For the first part, the computation results in \(-3\) and for the second part \(-14\). Understanding the multiplication of constants perfectly sets up the next phase of solving the equation.

Addition of Integers

The final phase of resolving the expression involves the addition of integers. After multiplying the constants, the results \(-3\) and \(-14\) are added together to generate the solution.
Adding integers, especially negative ones, can be tricky because of the sign rules.

• Think of adding negative numbers as combining debts; you simply sum their absolute values and retain the negative sign.
• In our calculation, \(-3\) added to \(-14\) equals \(-17\).

This step is essential as it finalizes the problem-solving process, yielding \(-17\) as the solution. Practice is key to getting comfortable with addition of integers, especially when they involve different signs.