Problem 56 - Prealgebra

Question

Solve each equation. $$x+19=32$$

Step-by-Step Solution

Answer

The solution is \( x = 13 \).

Step 1: Identify the Equation

The given equation is \( x + 19 = 32 \). Our goal is to solve for \( x \).

Step 2: Isolate the Variable

To isolate \( x \), we need to get it by itself on one side of the equation. To do this, subtract 19 from both sides of the equation: \( x + 19 - 19 = 32 - 19 \).

Step 3: Simplify the Equation

Simplifying both sides gives \( x = 13 \).

Step 4: Verify the Solution

To verify the solution, substitute \( x = 13 \) back into the original equation to check if the equality holds: \( 13 + 19 = 32 \). Since this is true, \( x = 13 \) is the correct solution.

Key Concepts

Variable Isolation

When we talk about variable isolation in an equation, we mean getting the variable (usually represented by a letter like \( x \)) alone on one side of the equation. In simpler terms, the variable should not have any other numbers or terms attached to it.
To isolate the variable in the equation \( x + 19 = 32 \), we need to remove the 19 from the left side where the \( x \) is located. This is done by performing the opposite operation of what is currently applied to \( x \). Here, 19 is added to \( x \), so we'll do the opposite, which is to subtract 19 from both sides of the equation.

• Original equation: \( x + 19 = 32 \)
• Subtracting 19 from both sides: \( x + 19 - 19 = 32 - 19 \)
• Simplified to: \( x = 13 \)

This step ensures the \( x \) is isolated, making it much easier to see its value.

Equation Verification

After isolating the variable and finding its value, it's crucial to verify that our solution is correct. Verification is essentially double-checking our work to ensure there are no mistakes.
To verify, we substitute the value of our isolated variable back into the original equation. For our equation \( x + 19 = 32 \), we found \( x = 13 \). We then plug \( x = 13 \) back into the original equation:

• Original equation with \( x = 13 \): \( 13 + 19 \)
• Calculate \( 13 + 19 \) to see if it equals 32.
• Since \( 13 + 19 = 32 \), our solution is verified as correct.

Verifying your solution can catch errors early on and gives confidence that the work is correct.

Arithmetic Operations

Arithmetic operations are the basic math operations such as addition, subtraction, multiplication, and division. They are essential tools used in solving equations like \( x + 19 = 32 \).
In our equation, we primarily employ subtraction, as our goal is to isolate the variable \( x \). Here's a breakdown of how subtraction played its role:

• Subtraction is used to both sides: \( x + 19 - 19 = 32 - 19 \).
• This simplifies the equation by eliminating 19 from the left side.
• We are left with \( x = 13 \).

These operations are fundamental in breaking down complex equations into simpler ones, making it feasible for us to find solutions with ease. It's crucial to apply them correctly to maintain the balance of equations.