Problem 11 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$x-4=19$$

Step-by-Step Solution

Answer

The solution of the equation \(x - 4 = 19\) is \(x = 23\).

Step 1: Isolate the variable x on one side

The equation is \(x - 4 = 19\). Add 4 to both sides of the equation. So it becomes \(x - 4 + 4 = 19 + 4\), which simplifies to \(x = 23\).

Step 2: Checking the solution

You can check this solution by substituting 23 for \(x\) in the original equation \(x - 4 = 19\). This gives \(23 - 4 = 19\), which is the original equation, and thus confirms that the solution is correct.

Key Concepts

Addition Property of Equality

The addition property of equality is a fundamental concept in solving linear equations. When you're working with an equation like \(x - 4 = 19\), this property allows you to add the same number to both sides of the equation without changing its balance. Imagine the equation as a balanced scale. Adding the same value to both sides ensures that the scale remains balanced:

• If we add \(4\) to both sides: \(x - 4 + 4 = 19 + 4\)
• The \(-4\) and \(+4\) cancel each other on the left, simplifying to \(x = 23\)

Using this property effectively helps maintain equality and simplify equations as you work toward finding the value of the variable.

Variable Isolation

Variable isolation is the process of manipulating an equation to get the variable, in this case, \(x\), all by itself on one side of the equation.The goal is to have \(x =anumber\), which gives the solution instantly. Here's how it works:

• In \(x - 4 = 19\), the step of adding \(4\) to both sides is a strategic move to eliminate the \(-4\), leaving \(x\) isolated: \(x = 23\).
• This simplification makes it easier to identify the value \(x\) must take for the equation to hold true.

Variable isolation is a key technique in algebra that sets the stage for efficiently solving equations and finding unknowns.

Solution Verification

Solution verification is crucial to ensure the proposed solution is indeed correct.After solving the equation, you'll substitute the value back into the original equation to check if it satisfies the equation:

• With \(x = 23\), substitute it back into \(x - 4 = 19\).
• The calculation \(23 - 4 = 19\) confirms that the left-hand side matches the right-hand side.

If the equation balances, the solution is verified as correct. This process acts as a double-check, reducing errors in calculations and confirming the solution's validity before finalizing it.

Algebraic Manipulation

Algebraic manipulation involves rearranging an equation to make it easier to solve. The steps involved can vary, but generally, they include actions like adding, subtracting, multiplying, or dividing both sides of an equation.In the example \(x - 4 = 19\), the algebraic manipulation was straightforward:

• Add \(4\) to both sides to cancel out the \(-4\).
• Simplify to find \(x = 23\).

Such manipulations are used not only to solve equations, but also to simplify complex algebraic expressions and solve more advanced problems.