Problem 31 - Algebra A Combined Function

Question

Solve. $$ \sqrt{3 x+4}=5 $$

Step-by-Step Solution

Answer

The solution is \( x = 7 \).

Step 1: Understand the problem

We need to solve the equation \( \sqrt{3x+4}=5 \). This involves finding a value of \( x \) that makes the equation true.

Step 2: Eliminate the square root

To eliminate the square root, square both sides of the equation. This gives us:\[(\sqrt{3x+4})^2 = 5^2\]Thus, the equation simplifies to:\[3x+4 = 25\]

Step 3: Solve for x

Now, solve the linear equation \( 3x+4 = 25 \). Start by isolating \( 3x \) by subtracting 4 from both sides:\[3x = 25 - 4\]Which simplifies to:\[3x = 21\]

Step 4: Divide both sides by 3

To find \( x \), divide both sides by 3:\[x = \frac{21}{3}\]Which simplifies to:\[x = 7\]

Step 5: Verify the solution

Substitute \( x = 7 \) back into the original equation to ensure it satisfies \( \sqrt{3x+4} = 5 \):\[\sqrt{3(7)+4} = \sqrt{21+4} = \sqrt{25} = 5\]Since both sides are equal, \( x = 7 \) is a correct solution.

Key Concepts

Linear Equations

Linear equations are mathematical statements involving variables, typically in the first degree, meaning they don't have any variables raised to a power higher than one. To solve a linear equation, you aim to isolate the variable on one side of the equation. The basic form of a linear equation is \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants.When working with a linear equation, you often:

• Combine like terms if necessary.
• Move constants to one side using addition or subtraction.
• Isolate the variable by dividing or multiplying.

In the context of our exercise, once the square root is eliminated and the equation simplified to \( 3x + 4 = 25 \), it becomes a straightforward linear equation. By subtracting 4 from both sides, we make it easier to solve for \( x \). Dividing by 3 then gives us the value of \( x \). This demonstrates how linear equations form a foundational part of many algebraic problems.

Square Roots

Square roots involve finding a number which, when multiplied by itself, gives the original number. The square root symbol, \( \sqrt{} \), represents this operation. In an equation, square roots can often complicate the process because they involve an additional step to simplify.In solving equations with square roots:

• Identify the square root on one side of the equation.
• Eliminate the square root by squaring both sides.

Squaring both sides of an equation effectively removes the square root, making the problem simpler, just like in our equation \( \sqrt{3x + 4} = 5 \). Once squared, it transforms to \( 3x + 4 = 25 \), which is much simpler to solve. Always remember, when squaring both sides, ensure the resulting equation remains balanced, as this keeps our solutions accurate.

Verification of Solutions

Verification is a crucial step in solving equations, ensuring that the solution satisfies the original equation. After finding a potential solution, substitute it back into the initial equation to check if it holds true.The verification process involves:

• Substituting the obtained solution into the original equation.
• Simplifying both sides to confirm equality.

For our exercise, substituting \( x = 7 \) into \( \sqrt{3x + 4} = 5 \) confirms our solution is correct, as both sides equal 5. This step is not only about confirming accuracy but also about developing confidence in the solution you've derived. It's a fundamental practice that ensures reliability, especially when working with more complex equations where errors are more prone.