Question

Find the value of each expression if \(x=3\) and \(y=-2\) \(|3 x+2 y|\)

Short answer

5

Step-by-step solution

Substitute the values

Start by substituting the given values of 3 for x and -2 for y into the expression The expression is |3x + 2y| Substitute x = 3 and y = -2 into the expression: |3(3) + 2(-2)|

Simplify inside the absolute value

Simplify the expression inside the absolute value sign: |3(3) + 2(-2)| = |9 + (-4)| = |9 - 4|

Calculate the result

Finish simplifying the expression inside the absolute value and compute the absolute value: |9 - 4| = |5| = 5

Key concepts

Substitution in Algebra

Substitution in algebra is a critical concept. We often need to replace variables with their given values to solve expressions. For instance, in the exercise, we substitute the values of \(x\) and \(y\) into the expression \(|3x + 2y|\).
Here is how it is done:
1. Replace \(x\) with 3 and \(y\) with -2.
2. The expression \(|3x + 2y|\) then becomes \(|3(3) + 2(-2)|\).
This step helps to transform the algebraic problem into a numerical one, making it easier to solve.

Simplifying Expressions

Simplifying expressions is a cornerstone of algebra. It involves performing arithmetic operations to reduce an expression to its simplest form.
For example, in the substituted expression \(|3(3) + 2(-2)|\), we first carry out the multiplication:
1. \(3 \times 3 = 9\)
2. \(2 \times -2 = -4\)
Now, we add these results together: \(9 + (-4)\) which simplifies to \(9 - 4\).
At this point, the expression inside the absolute value is simplified to \(5\). Breaking down each operation is crucial for clarity and accurate results.

Absolute Value Calculations

The absolute value of a number is its distance from zero on a number line, regardless of direction. In simpler terms, it is always a non-negative value.
In our problem, after simplifying the expression inside the absolute value, we get \(|9 - 4| = |5|\).
The absolute value of 5 is simply 5, because 5 units from zero is 5.
The final step: After calculating the inside expression, wrap up by determining the absolute value, giving us the result 5.
Absolute value ensures our final answer is always non-negative, simplifying and clarifying the result.