Problem 42 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ 7(2 y-x), \text { if } x=-1 \text { and } y=2 $$

Step-by-Step Solution

Answer

The value of the expression is 35.

Step 1: Understanding the Expression

The expression given is \(7(2y - x)\). We are asked to find its value when \(x = -1\) and \(y = 2\). This means we need to substitute \(-1\) for \(x\) and \(2\) for \(y\).

Step 2: Substitute the Values

We substitute \(x = -1\) and \(y = 2\) into the expression. This gives: \[ 7(2(2) - (-1)) \].

Step 3: Simplify Inside the Parentheses

Calculate inside the parentheses: \(2(2) = 4\) and \(-(-1) = 1\). So the expression becomes: \[7(4 + 1)\].

Step 4: Compute Parentheses Result

Add the results inside the parentheses: \(4 + 1 = 5\). Now the expression is: \[7 \times 5\].

Step 5: Calculate the Final Result

Perform the multiplication: \(7 \times 5 = 35\). So, the value of the expression is 35.

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra. It involves replacing variables in an equation with given values, making calculations straightforward.

• Begin by identifying the variables in your expression. Here, they are \(x\) and \(y\).
• Take the values provided for these variables. In our exercise, \(x = -1\) and \(y = 2\).
  
• Substitute these values into the expression: \(7(2y - x)\). This becomes \(7(2(2) - (-1))\).

Substitution helps simplify expressions by eliminating variables, letting you work directly with numbers. This process is particularly useful in solving complex algebraic expressions.

Simplifying Expressions

Simplifying expressions is crucial in algebra as it helps reduce complexity and makes calculations easier.First, focus on the expression inside parentheses: \(2y - x\). After substitution, this becomes \(2(2) - (-1)\).

• Calculate each term separately: \(2 \times 2 = 4\) and \(-(-1) = 1\).
• The expression inside becomes \(4 + 1\).

Now we have a simpler expression, \(7(5)\).
Simplifying not only makes calculations less prone to error but also ensures a clearer understanding of the expression structure. Always work inside-out in expressions, tackling parentheses first before moving outward.

Multiplication in Algebra

Multiplication is a key operation in algebra that combines quantities. In the context of our exercise, it appears as \(7 \times (result\ from\ parentheses)\).Once we've simplified the expression to \(7(5)\), all that remains is straightforward multiplication. This is done by multiplying the number outside the parentheses (7) by the sum inside (5), resulting in \(7 \times 5 = 35\).

• Ensure every number in the product is correct by double-checking preceding steps.
• Perform the multiplication step with care to avoid mistakes.

Multiplication helps in efficiently evaluating expressions, especially when dealing with repeated addition or scaling values. Mastering this concept is vital as it forms the basis of many algebraic operations.