Chapter 13: Problem 4
Solve for \(x\) and check. $$\sqrt{3 x-2}=5$$
Short Answer
Expert verified
The solution is 9.
Step by step solution
01
Isolate the Square Root
Begin by isolating the square root on one side of the equation. In this case, the square root is already isolated on the left side.
02
Square Both Sides of the Equation
Remove the square root by squaring both sides of the equation: \((\sqrt{3x-2})^2 = 5^2\).
03
Simplify the Squared Terms
Squaring the square root cancels it out on the left side, and squaring 5 gives you 25. You are left with the equation \(3x - 2 = 25\).
04
Add 2 to Both Sides
Isolate the term with the variable by adding 2 to both sides of the equation, giving you \(3x = 27\).
05
Divide Both Sides by 3
Solve for x by dividing both sides of the equation by 3, giving you \(x = \frac{27}{3}\).
06
Simplify the Fraction
Simplify the fraction \(\frac{27}{3}\) to get the value of x, which is 9.
07
Check the Solution
Substitute x with 9 back into the original equation \(\sqrt{3(9)-2}=5\). Simplify the expression inside the square root to find that it equals 25. Since the square root of 25 is indeed 5, the solution checks out.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mathematical Problem-Solving
Problem-solving in mathematics is a key skill that empowers you to tackle various challenges by applying logical steps. The journey to solve for a variable, like the equation \(\sqrt{3x-2}=5\), requires methodical steps that ensure precision and accuracy. Initially, understanding the problem lays the foundation. Here, identifying that you have a square root equation sets the stage for the solution process.
Next is devising a plan by outlining the steps you shall take, such as isolating the square root, performing algebraic operations, and simplifying where necessary. After executing the plan, as shown in the provided steps, you reach a potential solution. The final phase is critical: looking back to reassess the solution through checking. This holistic approach not only builds your mathematical skills but also enhances critical thinking that applies beyond the realm of mathematics.
Next is devising a plan by outlining the steps you shall take, such as isolating the square root, performing algebraic operations, and simplifying where necessary. After executing the plan, as shown in the provided steps, you reach a potential solution. The final phase is critical: looking back to reassess the solution through checking. This holistic approach not only builds your mathematical skills but also enhances critical thinking that applies beyond the realm of mathematics.
Isolate the Square Root
When faced with square root equations, it’s essential to isolate the square root before proceeding with other operations. This involves making sure that the square root expression is alone on one side of the equation, which allows for clearer steps moving forward.
In the example \(\sqrt{3x-2}=5\), the square root is already on one side, meaning this crucial first step is complete without further manipulation. Isolation of the square root is a step that simplifies the problem and prepares it for easier manipulation. Notably, if the square root were not alone, you would need to perform algebraic operations to move other terms to the opposite side of the equation.
In the example \(\sqrt{3x-2}=5\), the square root is already on one side, meaning this crucial first step is complete without further manipulation. Isolation of the square root is a step that simplifies the problem and prepares it for easier manipulation. Notably, if the square root were not alone, you would need to perform algebraic operations to move other terms to the opposite side of the equation.
Simplify Equations
Simplifying equations is the heart of algebraic problem-solving. It involves breaking down complex expressions into more manageable forms. This process helps to clarify the necessary operations and brings you closer to finding the variable’s value. After squaring both sides of our isolated square root equation, \(3x - 2 = 25\), the next steps involve simplifying the algebraic expression.
You do this by performing inverse operations. In this example, adding 2 to both sides neutralizes the minus 2, simplifying the equation further to \(3x = 27\). Division by the coefficient of the variable, which is 3 in this case, finally simplifies the equation to \(x = 9\). Through these steps, not only is the equation simplified, but the solution is made apparent.
You do this by performing inverse operations. In this example, adding 2 to both sides neutralizes the minus 2, simplifying the equation further to \(3x = 27\). Division by the coefficient of the variable, which is 3 in this case, finally simplifies the equation to \(x = 9\). Through these steps, not only is the equation simplified, but the solution is made apparent.
Check Mathematical Solutions
Once you've calculated an answer, checking the solution confirms its validity and ensures that no errors occurred during the problem-solving process. To check the solution to the square root equation \(\sqrt{3x-2}=5\), we substitute our found value of \(x\) back into the equation.
For the solution \(x = 9\), we plug it into the original equation: \(\sqrt{3(9)-2}\). Simplifying inside the square root, we get \(\sqrt{27-2}=\sqrt{25}\), which indeed equals 5. This verification step assures us that the solution is correct, bringing the problem-solving process to a successful conclusion. It reinforces the understanding of mathematical equations and builds confidence in the ability to solve similar problems in the future.
For the solution \(x = 9\), we plug it into the original equation: \(\sqrt{3(9)-2}\). Simplifying inside the square root, we get \(\sqrt{27-2}=\sqrt{25}\), which indeed equals 5. This verification step assures us that the solution is correct, bringing the problem-solving process to a successful conclusion. It reinforces the understanding of mathematical equations and builds confidence in the ability to solve similar problems in the future.
