Problem 62 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=-5, y=4,\) and \(t=10 .\) See Example 6. $$ |x+t-7 y| $$

Step-by-Step Solution

Answer

The value of the expression is 23.

Step 1: Substitute the Values

Start by substituting the given values into the expression. The expression is \(|x + t - 7y|\). We know \(x = -5\), \(t = 10\), and \(y = 4\). This gives us \(|-5 + 10 - 7(4)|\).

Step 2: Simplify Inside the Absolute Value

Now, perform the simplification inside the absolute value. Calculate \(-5 + 10\) which equals \(5\). Then, calculate \(7 \times 4 = 28\). Substitute these results back into the expression to get \(|5 - 28|\).

Step 3: Evaluate the Expression

Subtract 28 from 5: \(5 - 28 = -23\). So, the expression becomes \(|-23|\).

Step 4: Apply the Absolute Value

The absolute value operation turns any negative number into its positive counterpart. Therefore, \(|-23| = 23\).

Key Concepts

Understanding Absolute Value

The concept of absolute value is a fundamental part of algebra that often finds its way into various mathematical problems. Absolute value refers to the distance a number is from zero on the number line, without considering its direction. This means absolute values are always non-negative. For example:

• The absolute value of \(-23\) is \(|-23| = 23\).
• The absolute value of 5 is \( |5| = 5 \).

In the given problem, after substituting and simplifying the expression, we arrived at \(|-23|\). By applying the absolute value operation, we convert \-23\ into 23, as the absolute value only concerns the magnitude of a number. This concept helps in simplifying problems that deal with distances or differences.

Expression Substitution

Expression substitution is the process of replacing variables in an expression with specific values. This operation simplifies the evaluation process, making algebraic expressions easier to handle. In our exercise:

• We are given the values: \(x = -5\), \(y = 4\), and \(t = 10\).
• The expression to evaluate is \(|x + t - 7y|\).
• By substituting the given values, the expression transforms into \(|-5 + 10 - 7(4)|\).

This substitution step is crucial as it prepares the expression for further simplification by replacing abstract variables with concrete numbers. By doing so, we can focus on numerical computation rather than symbolic manipulation, which is often easier for learners to understand.

Expression Simplification

Expression simplification involves performing operations within an algebraic expression to reduce it to its simplest form. After substitution, simplifying the expression helps us understand the calculation more clearly. In our problem:

• We start by simplifying inside the absolute value: \(-5 + 10\) results in \(+5\).
• Next, we calculate \(7 \times 4 = 28\).
• We substitute these back, leading to \(5 - 28\).

The simplification process involves arithmetic within the parentheses and other operations such as multiplication and subtraction. These basic mathematical operations bring us closer to solving the expression entirely by reducing it to a manageable form.

Performing Mathematical Operations

Understanding the sequence and type of operations is essential in solving any algebraic expression. In our exercise, several basic mathematical operations were used:

• **Addition**: Calculating \(-5 + 10\) to get \(5\).
• **Multiplication**: \(7 \times 4 = 28\).
• **Subtraction**: From \(5 - 28\), we get \(-23\).

These operations follow the standard order of operations, which is crucial in algebra. By performing operations in the correct sequence, we maintain the integrity of the expression's transformation from start to finish. Mistakes in order can lead to incorrect results, so understanding this hierarchy is key. Computation practice in this sequence cements foundational arithmetic skills, useful for tackling more complex problems.