Problem 399 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{6 v-2}-10=0\)

Step-by-Step Solution

Answer

\(v = 17\)

Step 1: Isolate the square root

Add 10 to both sides of the equation to isolate the square root term: o \(\sqrt{6v - 2} - 10 = 0 \)o becomes o \(\sqrt{6v - 2} = 10 \)..

Step 2: Square both sides

Square both sides of the equation to remove the square root: o \((\sqrt{6v - 2})^2 = 10^2 \)o becomes o \(6v - 2 = 100 \)..

Step 3: Solve for v

Add 2 to both sides of the equation to isolate the term containing v: o \(6v - 2 + 2 = 100 + 2 \)o becomes o \(6v = 102 \).Finally, divide by 6 to solve for v: o \(v = \frac{102}{6} \)o which simplifies to o \(v = 17 \)..

Key Concepts

square root

The square root is a mathematical function that, when applied to a number, returns the value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 * 5 = 25. Notating square roots in math is typically done using the radical symbol \(\backslashsqrt{}\). When solving equations involving square roots, it is crucial to isolate the square root term to simplify the process.

isolating variables

Isolating variables means manipulating an equation so the variable of interest stands alone on one side. This often involves a series of algebraic steps such as addition, subtraction, multiplication, or division. For example, in the equation \(\backslashsqrt{6v - 2} - 10 = 0\), we add 10 to both sides to start isolating the variable: \(\backslashsqrt{6v - 2} = 10\). By performing inverse operations, we simplify the equation step by step until the variable is by itself.

squaring both sides

Squaring both sides of an equation is a method used to eliminate a square root. When dealing with square roots, it is often essential to square both sides to transform the equation into a simpler form. For instance, if you have \(\backslashsqrt{6v - 2} = 10\), squaring both sides gives \(( \backslashsqrt{6v - 2})^2 = 10^2\). This simplifies to \(6v - 2 = 100\), making it easier to solve for the variable.

addition and division in equations

Addition and division are core techniques in solving equations. For instance, after squaring both sides of \(\backslashsqrt{6v - 2} = 10\) to get \(6v - 2 = 100\), we need to use addition and division to isolate and solve for v. First, add 2 to both sides: \(6v - 2 + 2 = 100 + 2\) or simply, \(6v = 102\). Then, divide by 6: \(v = \backslashfrac{102}{6}\), which simplifies to \(v = 17\). Each step helps in methodically breaking down the problem to find the solution.