Problem 77 - Elementary Algebra

Question

For the following problems, solve the equations. $$ \sqrt{5 b+4}-5=-2 $$

Step-by-Step Solution

Answer

Answer: The value of b in the given equation is 1.

Step 1: Isolate the square root term

Add 5 to both sides of the equation to isolate the square root term. $$ \sqrt{5b + 4} - 5 + 5 = -2 + 5 $$ $$ \sqrt{5b + 4} = 3 $$

Step 2: Square both sides of the equation

Square both sides of the equation to eliminate the square root sign. $$ (\sqrt{5b + 4})^2 = 3^2 $$ $$ 5b + 4 = 9 $$

Step 3: Isolate the variable b

Subtract 4 from both sides of the equation and then divide both sides by 5 to isolate b. $$ 5b + 4 - 4 = 9 - 4 $$ $$ 5b = 5 $$ $$ b = \frac{5}{5} $$ $$ b = 1 $$ The solution to the equation is \(b = 1\).

Key Concepts

Isolate the Variable

One initial step in solving square root equations, as well as many other types of equations, is to isolate the variable. This process involves moving all the terms without the variable on one side of the equation so that the variable term is by itself on the other side. In our example, we start with the equation \( \sqrt{5b + 4} - 5 = -2 \). To isolate the square root term—which contains our variable b—we add 5 to both sides of the equation. This gives us \( \sqrt{5b + 4} = 3 \), effectively isolating the square root on one side.

Isolating the variable is a cornerstone of algebra and has a wide array of applications. It’s crucial for simplifying complex equations and setting the stage for other operations, like squaring both sides, that you might need to perform to solve the equation.

Square Both Sides

In an equation with a square root, a common technique to solve for the variable is to square both sides of the equation. When you square the square root of a number, you are essentially undoing the square root and are left with just the number itself. Looking again at the isolated square root \( \sqrt{5b + 4} = 3 \), by squaring both sides, we get \( (\sqrt{5b + 4})^2 = 3^2 \), and then \( 5b + 4 = 9 \).

Squaring both sides is not just limited to square root equations—it's a technique that can be used in any circumstance where it helps to remove exponents or radicals. This step is particularly powerful because it helps us transform an equation into something more straightforward and solvable through basic algebraic operations.

Square Root Properties

Understanding square root properties is essential when working with square root equations. The square root of a number x is a number that, when multiplied by itself, gives x. The primary property we use in our example is the fact that squaring a square root cancels the radical, leaving the argument of the square root unchanged \( (\sqrt{x})^2 = x \). Another property to be aware of is that the square root of a product is equal to the product of the square roots of the factors \( \sqrt{xy} = \sqrt{x} \times \sqrt{y} \).

Using these properties properly ensures that we are manipulating equations correctly and helps prevent errors that could lead to misunderstanding the solutions to problems involving square roots.

Elementary Algebra Equations

The manipulation of elementary algebra equations is a fundamental skill in mathematics. This involves operations like adding, subtracting, multiplying, and dividing amounts from both sides in order to find the value of the unknown variable. In the final steps of our example, we simplify \( 5b + 4 = 9 \) by subtracting 4 from both sides, which gives us \( 5b = 5 \). Dividing both sides by 5 gives us \( b = \frac{5}{5} \), and simplifying that, we find \( b = 1\).

Elementary algebra equations are not only foundational for higher mathematics but also have countless applications in science, engineering, and beyond. Mastery of solving such equations is not just about getting the correct answer—it’s about understanding the relationship between numbers and operations.