Solve for x (2 square root of x)/3+1=7
You are presented with an algebraic equation where you have to find the value of the variable 'x'. The equation includes a fractional term with a numerator that is two times the square root of 'x' and a denominator of three. This fraction is then increased by one and set equal to seven. Your task is to isolate 'x' and determine its value through a series of algebraic steps that may include inverse operations such as subtraction, multiplication, square, and division.
$\frac{2 \sqrt{x}}{3} + 1 = 7$
Answer
Solution:
Step 1: Isolate $2 \sqrt{x}$.
Step 1.1: Move terms without $2 \sqrt{x}$ to the opposite side.
Step 1.1.1: Subtract $1$ from both sides to get $\frac{2 \sqrt{x}}{3} = 7 - 1$.
Step 1.1.2: Perform the subtraction to find $\frac{2 \sqrt{x}}{3} = 6$.
Step 1.2: Eliminate the denominator by multiplying by $3$ to obtain $3 \cdot \frac{2 \sqrt{x}}{3} = 3 \cdot 6$.
Step 1.3: Simplify both sides.
Step 1.3.1: Focus on the left side.
Step 1.3.1.1: Remove the common factor of $3$.
Step 1.3.1.1.1: Apply cancellation to get $2 \sqrt{x} = 3 \cdot 6$.
Step 1.3.2: Now simplify the right side.
- Step 1.3.2.1: Multiply $3$ and $6$ to find $2 \sqrt{x} = 18$.
Step 2: Square both sides to eliminate the square root.
- Square the equation to get $(2 \sqrt{x})^2 = 18^2$.
Step 3: Simplify the squared equation.
Step 3.1: Express $\sqrt{x}$ as $x^{\frac{1}{2}}$.
Step 3.2: Simplify the left side.
Step 3.2.1: Apply the power rule to $(2 x^{\frac{1}{2}})^2$.
Step 3.2.1.1: Square the coefficient $2$ to get $4 (x^{\frac{1}{2}})^2 = 18^2$.
Step 3.2.1.2: Apply the exponent rule to get $4 x^{1} = 18^2$.
Step 3.3: Simplify the right side to get $4 x = 324$.
Step 4: Solve for $x$.
Step 4.1: Divide the equation $4 x = 324$ by $4$.
Step 4.2: Simplify the left side to isolate $x$.
- Step 4.2.1: Cancel out the $4$ to get $x = \frac{324}{4}$.
Step 4.3: Simplify the right side to find the value of $x$.
- Step 4.3.1: Divide $324$ by $4$ to get $x = 81$.
Knowledge Notes:
The problem-solving process involves algebraic manipulation to isolate and solve for the variable $x$. Here are the relevant knowledge points:
Isolating a Variable: This involves moving terms around in an equation to get the variable of interest by itself on one side of the equation. This often requires adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Simplifying Expressions: This involves combining like terms and reducing expressions to their simplest form. It can include canceling out common factors and performing arithmetic operations.
Square Roots and Exponents: The square root of a number $x$ is written as $\sqrt{x}$ and is equivalent to $x^{1/2}$. Squaring both sides of an equation is a common technique used to eliminate square roots.
The Power Rule: This rule states that $(a^m)^n = a^{m \cdot n}$, which is used when raising a power to another power.
Solving Quadratic Equations: When one side of an equation is squared, it often leads to a quadratic equation. However, in this case, once the square root is eliminated, the equation becomes linear.
Division and Simplification: The final step in solving for $x$ typically involves dividing both sides of the equation by a number to isolate $x$ and then simplifying the result to find the solution.
