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Chapter 2: Problem 47

Solve each equation. Check each solution. $$ 6 x+10=-20 $$

Short Answer

Expert verified
The solution is \(x = -5\).

Step by step solution

01

Isolate the variable term

Start with the equation: \[ 6x + 10 = -20 \]To isolate the term with the variable (\(6x\)), subtract 10 from both sides of the equation:\[ 6x + 10 - 10 = -20 - 10 \]This simplifies to:\[ 6x = -30 \]
02

Solve for the variable

Now that you have \(6x = -30\), you need to solve for \(x\). Divide both sides of the equation by 6:\[ \frac{6x}{6} = \frac{-30}{6} \]This simplifies to:\[ x = -5 \]
03

Check the solution

Substitute \(x = -5\) back into the original equation \(6x + 10 = -20\) to verify the solution:\[ 6(-5) + 10 = -20 \]Calculate:\[ -30 + 10 = -20 \]Since \(-20 = -20\), the solution is correct.

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Key Concepts

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

Solving equations
Solving equations is about finding the value of the variable that makes the equation true. Equations are like statements that assert two expressions are equal. In the given exercise, our task is to determine the value of \(x\) that satisfies the equation:
  • Start with the entire equation without altering its balance.
  • Apply operations to both sides to maintain equality.
  • Aim to simplify the equation step by step, focusing on the variable.
In this problem, by performing operations like subtracting a constant (10) from both sides, we make progresses toward isolating the variable term \(6x\), simplifying the path to finding \(x\). In general, while solving equations, using addition or subtraction gently changes the equation towards an easier form. Multiplication or division will then help in splitting or uniting terms, eventually making the equation less complicated.
Isolation of variables
Isolation of variables is a crucial step in solving equations where the goal is to have the variable by itself on one side of the equation. This involves moving other terms to the opposite side, essentially making it clear what the variable represents numerically. In our example:
  • We begin with the equation \(6x + 10 = -20\).
  • By subtracting 10 from each side, we isolate the term \(6x\).
  • This rearranges the equation to \(6x = -30\).
After isolating the term \(6x\), we proceed to unravel it by dividing by the coefficient (6) to completely isolate \(x\). This results in the finalized equation \(x = -5\). Isolation is like unwrapping a package containing only what's necessary to see, all other superfluous parts having been methodically peeled away.
Checking solutions
Checking solutions involves plugging the obtained value back into the original equation to either confirm or negate its correctness. This step assures the result derived is indeed accurate and fulfills the equation's requirements.
  • Take the solution \(x = -5\).
  • Substitute back into the original equation: \(6(-5) + 10\).
  • Calculate to verify: \(-30 + 10 = -20\).
Upon checking, the verification matches the original equation, meaning the solution satisfies the initial condition \(6x + 10 = -20\). This step is vital, as errors may arise from miscalculations or missteps amidst isolation. Checking is similar to proofreading; it assures the correctness and reliability of your result.

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