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Chapter 2: Problem 1

What would you do first when solving each equation? $$6 x-9=-15$$

Short Answer

Expert verified
The first action is to add 9 on both sides of the equation to isolate the term with \(x\) on one side, and the final value of \(x\) is \(-1\).

Step by step solution

01

Isolate the variable term

In order to isolate \( x \) on one side of the equation, add 9 on both sides of the equation. This will cancel out the negative 9 with the 9 that are added on the left side, thus: \( 6x - 9 + 9 = -15 + 9 \). This simplifies to \( 6x = - 6 \)
02

Solve for x

To solve for \( x \), divide both sides of the equation by the coefficient of \( x \) which is 6: \[ 6x / 6 = -6 / 6 \]. This would give the value of \( x \) as \(-1\)

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Key Concepts

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

Isolating the Variable
When you are faced with a linear equation like \(6x - 9 = -15\), your first goal is to isolate the variable. This means you need to get \(x\) by itself on one side of the equation. Why is this important? Because isolating \(x\) simplifies the equation, making it easier to solve.

To achieve this, we start by getting rid of any constants and other terms on the same side as \(x\). In our example, \(6x - 9 = -15\) has a \(-9\) that needs to be moved to the other side of the equation. You can do this by performing inverse operations, like adding 9 to both sides:
  • Add 9 to both the left and right sides: \(6x - 9 + 9 = -15 + 9\)
  • This simplifies to \(6x = -6\)
Once this step is complete, you have isolated \(x\) on one side, ready for the next step.
Solving for x
Once you've successfully isolated the variable \(x\), the next task is solving for it. This is about finding the value that makes the equation true.

With the equation simplified to \(6x = -6\), your goal is to eliminate the coefficient of \(x\), which in this instance is 6. You can achieve this by dividing each side of the equation by 6. Remember:
  • Divide both sides by 6: \(\frac{6x}{6} = \frac{-6}{6}\)
  • You'll then find that \(x = -1\)
This process is often referred to as performing operations that "undo" what has been done to \(x\). It ensures that you find the exact value of \(x\) that satisfies the original equation.
Linear Equations
Linear equations are mathematical statements that describe how two expressions are equal, where the highest power of the variable is one. A typical example might look like \(ax + b = c\). In these equations, the graph represents a straight line.

Linear equations are common in algebra and serve as foundational skills for more complex math topics. They follow a predictable set of steps that can be applied to solve them.
  • Identify the variable term and constants.
  • Use inverse operations to isolate the variable.
  • Simplify and solve the equation step-by-step.
Understanding linear equations is crucial as they often occur in various real-life applications. With practice, solving them becomes intuitive, aiding in honing your algebraic skills for more advanced math challenges.

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