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Chapter 7: Problem 10

Solve. $$ \sqrt{7 x-3}=5 $$

Short Answer

Expert verified
x = 4

Step by step solution

01

Isolate the Square Root

The square root is already isolated on the left side of the equation: \[ \sqrt{7x-3} = 5 \]
02

Square Both Sides

To eliminate the square root, square both sides of the equation: \[ (\sqrt{7x-3})^2 = 5^2 \] which simplifies to: \[ 7x - 3 = 25 \]
03

Solve for x

Now solve for \(x\):1. Add 3 to both sides to isolate the term with \(x\): \[ 7x - 3 + 3 = 25 + 3 \] which simplifies to: \[ 7x = 28 \]2. Divide both sides by 7 to solve for \(x\): \[ x = \frac{28}{7} \] which simplifies to: \[ x = 4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

isolating the variable
When solving square root equations, a crucial first step is to isolate the variable inside the square root. Isolating the variable means we want to get the expression with the variable by itself on one side of the equation.

In the given exercise, this step is already done for us. The equation \(\sqrt{7x-3}=5\) has the square root isolated on the left side. This simplifies our task because we can proceed directly to the next step without any further manipulation.

However, in other equations, you might need to move other terms around using addition or subtraction before isolating the square root term.
squaring both sides
To solve for the variable inside a square root, the square root must be eliminated first. We achieve this by squaring both sides of the equation.

In the given problem: \(\sqrt{7x-3}=5\)
we square both sides:
\[ (\sqrt{7x-3})^2 = 5^2 \]
This simplifies to:
\[ 7x - 3 = 25 \]
Squaring both sides removes the square root, leaving us with a simpler linear equation to solve.
simplification steps
Simplification is about making the equation easier to solve step by step.

After squaring both sides, we got:
\[ 7x - 3 = 25 \]
To isolate the variable term, you perform basic arithmetic operations:
  • Step 1: Add 3 to both sides to eliminate the constant term beside \(7x\)
    \[ 7x - 3 + 3 = 25 + 3 \]
    which simplifies to:
    \[ 7x = 28 \]
  • Step 2: Divide both sides by 7 to solve for \(x\):
    \[ x = \frac{28}{7} \] which simplifies to:
    \[ x = 4 \]
solving linear equations
The final stage of solving a square root equation involves solving the resulting linear equation. A linear equation is an equation in the form of \(ax + b = c\).

In our simplified equation:
\[ 7x = 28 \]
we solve for \(x\) by isolating it.

Here's the breakdown:
  • Step 1: We divide both sides by 7:
    \[ x = \frac{28}{7} \]
    which simplifies to:
    \[ x = 4 \]
That's the solution for \(x\). Always remember to check your answer by substituting \(x = 4\) back into the original equation to ensure it satisfies the equation.

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Most popular questions from this chapter

Simplify. $$ 7 x \sqrt{(x+y)^{3}}-5 x y \sqrt{x+y}-2 y \sqrt{(x+y)^{3}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt{b^{3}} \sqrt[5]{b^{4}} $$

The absolute value of a complex number \(a+b i\) is its distance from the origin. (See the graph above.) Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}}\) Find the absolute value of each complex number. $$|8-6 i|$$

Find the midpoint of the segment with the given endpoints. $$ (2,-1) \text { and }(5,8) $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \frac{\sqrt[5]{x^{3} y^{4}}}{\sqrt{x y}} $$
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