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Chapter 12: Problem 77

To prepare for Section 12.6, review solving equations. Solve. $$ 17 x-15=0 $$

Short Answer

Expert verified
The solution is \( x = \frac{15}{17} \).

Step by step solution

01

- Isolate the Variable

To isolate the variable, move the constant term to the other side of the equation. Start by adding 15 to both sides of the equation.\[ 17x - 15 + 15 = 0 + 15 \]This simplifies to:\[ 17x = 15 \]
02

- Solve for the Variable

Now, isolate the variable by dividing both sides of the equation by the coefficient of \(x\).\[ \frac{17x}{17} = \frac{15}{17} \]This simplifies to:\[ x = \frac{15}{17} \]

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Key Concepts

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

Isolate the Variable
When solving linear equations, our first goal is to isolate the variable on one side of the equation. In the given problem, we start with the equation:

17x-15=0.

To isolate the variable, we need to move the constant term (-15) to the other side of the equation. We do this by doing the opposite operation, which is adding 15 to both sides. This gives us:

\( 17x - 15 + 15 = 0 + 15 \)
This simplifies to:

\( 17x = 15 \)

Now we have the variable term isolated on one side of the equation, which is necessary before solving for the variable itself.
Linear Equations
A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. The equation in our exercise:

\( 17x-15=0 \)

is a classic example of a linear equation. Here are some properties of linear equations:


  • They form a straight line when graphed on a coordinate plane.
  • They have only one variable raised to the first power (no exponents other than 1).
  • They can have one solution, no solution, or infinitely many solutions, but in simple equations like ours, there is usually one unique solution.

Understanding the nature of linear equations helps us follow specific steps to find the solution efficiently.
Solving Equations
Once the variable is isolated, the next step is to solve the equation. In our problem, we have:

\( 17x = 15 \)
to solve for \( x \), we need to divide both sides by the coefficient of \( x \), which is 17:

\( \frac{17x}{17} = \frac{15}{17} \)

This simplifies to:

\( x = \frac{15}{17} \)

We've now solved the equation! The solution to the linear equation \( 17x-15=0 \) is \( x = \frac{15}{17} \). Remember, each step must balance the equation while maintaining equality for both sides.

Solving equations involves systematic approaches that include:

  • Isolating the variable
  • Simplifying the equation
  • Performing inverse operations
  • Checking the solution by substituting back into the original equation

Following these steps ensures you correctly solve linear equations every time.

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