Problem 429 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{9 p+9}=\sqrt{10 p-6}\)

Step-by-Step Solution

Answer

The solution is \( p = 15 \).

Step 1 - Eliminate the square root

Square both sides of the equation to eliminate the square roots. \ \ \ \[ \left( \sqrt{9p + 9} \right)^2 = \left( \sqrt{10p - 6} \right)^2 \] \ \[ 9p + 9 = 10p - 6 \]

Step 2 - Rearrange the equation

Subtract \(9p\) from both sides to isolate the variable term on one side: \ \ \[ 9p + 9 - 9p = 10p - 6 - 9p \] \ \[ 9 = p - 6 \]

Step 3 - Solve for the variable

Add 6 to both sides to solve for \(p\): \ \ \[ 9 + 6 = p - 6 + 6 \] \ \[ 15 = p \]

Step 4 - Verify the solution

Substitute \(p = 15\) back into the original equation to verify: \ \[ \sqrt{9(15) + 9} = \sqrt{10(15) - 6} \] \ \[ \sqrt{144} = \sqrt{144} \] \ \[ 12 = 12 \] The solution is verified.

Key Concepts

solving equations

Solving equations is a fundamental part of algebra. This process involves finding the value of the variable that makes the equation true. Usually, we isolate the variable on one side of the equation by performing operations such as addition, subtraction, multiplication, or division. For example, in our exercise, we need to solve the equation \(\sqrt{9p+9}=\sqrt{10p-6}\). First, we eliminate the square roots by squaring both sides. This gives us \[9p+9 = 10p - 6\]. The next steps involve simplifying and rearranging the equation to isolate the variable, leading us to find the solution.

square roots

Square roots are a mathematical concept where a number, when multiplied by itself, gives the original number under the square root. For instance, in \(\sqrt{144}\), since \(12\times12=144\), \(\sqrt{144} = 12\). In algebra, square roots often appear in equations that need solving. To solve our problem, we first had square roots on both sides of the equation. To eliminate them, we squared both sides because \(\left(\sqrt{x}\right)^2=x\). This step simplified our equation and allowed us to focus on solving the linear equation that remained.

algebraic manipulation

Algebraic manipulation involves rearranging and simplifying equations to find the value of the unknown variable. This might include operations like adding, subtracting, multiplying, or dividing both sides of the equation by the same amount. In our example, after squaring both sides, we had \[9p+9=10p-6\]. We then subtracted \(9p\) from both sides to get \[9=p-6\]. Afterward, adding 6 to both sides leads us to \[15=p\]. This step-by-step transformation simplifies the equation and isolates the variable, making it easier to find the solution.

verification in algebra

Verification in algebra is the process of substituting the found solution back into the original equation to ensure it is correct. This ensures that our solution satisfies the original problem. In the given problem, after finding \(p=15\), we substitute it back into the original equation: \[\sqrt{9(15)+9}=\sqrt{10(15)-6}\]. Simplifying both expressions, we get \[\sqrt{144}=\sqrt{144}\], which verifies that \(12=12\). Therefore, the solution is correct. Verification helps in checking the accuracy of the solution and is a crucial step in problem-solving.