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Chapter 8: Problem 95

Solve each equation. $$ 6 x+13=0 $$

Short Answer

Expert verified
The solution is \( x = -\frac{13}{6} \).

Step by step solution

01

- Isolate the variable term

Begin by isolating the term with the variable. To do this, subtract 13 from both sides of the equation:\[ 6x + 13 - 13 = 0 - 13 \]This simplifies to:\[ 6x = -13 \]
02

- Solve for the variable

Next, solve for the variable by dividing both sides of the equation by 6:\[ \frac{6x}{6} = \frac{-13}{6} \]This simplifies to:\[ x = -\frac{13}{6} \]

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Key Concepts

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

headline of the respective core concept
When solving linear equations, one of the main goals is to isolate the variable. In our given problem, we started with the equation:
  • 6x + 13 = 0
To isolate the variable term, we need to get rid of any constants on the same side of the equation as the variable. Here, the constant is 13. To remove it, we subtract 13 from both sides of the equation:
  • 6x + 13 - 13 = 0 - 13
After doing this subtraction, we're left with:
  • 6x = -13
Now the variable term is isolated, making it easier to solve for the variable in the next step.
headline of the respective core concept
Linear equations are a fundamental concept in algebra. They represent a relationship between variables where the highest power of the variable is 1. The general form of a linear equation is:
  • ax + b = c
Here, 'a', 'b', and 'c' are constants, and 'x' is the variable. In our problem, the equation fits this form:
  • 6x + 13 = 0
Understanding linear equations helps in solving for the variable by manipulating the equation and applying inverse operations. In this case, we first handle addition and subtraction before moving to division.
headline of the respective core concept
Algebraic manipulation is the process of rearranging and simplifying equations using algebraic properties. In solving our equation, we used subtraction and division, two common algebraic operations.To recap, we subtracted 13 from both sides:
  • 6x + 13 - 13 = 0 - 13
This simplified to:
  • 6x = -13
Next, we divided both sides by 6 to solve for x:
  • \[\frac{6x}{6} = \frac{-13}{6}\]
Simplifying this, we found:
  • x = -\frac{13}{6}
Each step involves basic algebraic manipulation, crucial for solving more complex equations.
headline of the respective core concept
Solving for x is the ultimate goal when working with equations. After isolating the variable, we need to solve for x by performing the appropriate algebraic operations. In our example:
  • 6x = -13
We see that x is multiplied by 6. To solve for x, we divide both sides by 6:
  • \[ \frac{6x}{6} = \frac{-13}{6} \]
This simplifies to:
  • x = -\frac{13}{6}
Now we have the solution for x. This process of solving for x involves inverse operations, which cancel out the current operations applied to the variable.

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