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Chapter 11: Problem 41

Solve each equation. Be sure to check each result. $$ 7 x+3 x=0 $$

Short Answer

Expert verified
The solution is \(x = 0\).

Step by step solution

01

Combine Like Terms

Start by combining the like terms on the left side of the equation. The equation is \(7x + 3x = 0\). Combine the terms to have \((7 + 3)x = 0\), which simplifies to \(10x = 0\).
02

Solve for x

Now that we have the equation \(10x = 0\), isolate \(x\) by dividing both sides of the equation by 10. This gives \(x = \frac{0}{10}\). Simplifying this expression, we find \(x = 0\).
03

Check the Solution

Substitute \(x = 0\) back into the original equation \(7x + 3x = 0\) to ensure that it satisfies the equation. Substitute to have \(7(0) + 3(0) = 0\), which simplifies to \(0 = 0\). Since both sides of the equation are equal, \(x = 0\) is the correct solution.

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Key Concepts

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

Combining Like Terms
In the realm of algebra, solving equations often starts with the process of combining like terms. Like terms are terms that have the same variable raised to the same power. For example, in the equation \(7x + 3x = 0\), the terms \(7x\) and \(3x\) are considered like terms because they both consist of the variable \(x\).
  • Combining these terms involves adding their coefficients together.
  • The coefficient is the numerical value in front of the variable.

In our equation, we add the numerical coefficients 7 and 3 to get 10. Therefore, \(7x + 3x\) simplifies to \(10x\).
This step of combining makes the equation simpler and easier to solve. Remember, always look for terms that you can combine at the outset of solving any equation.
Isolating Variable
After combining like terms, the next step is to isolate the variable. Isolating a variable means getting it all by itself on one side of the equation. This helps us find the solution to the equation.
In the equation \(10x = 0\), to isolate \(x\), we divide both sides by 10, which is the coefficient of \(x\).
  • The reason for dividing by 10 is to cancel out the coefficient and leave \(x\) on its own.
  • This operation must balance both sides of the equation to maintain equality.

Performing this division gives us \(x = \frac{0}{10}\). Simplifying \(\frac{0}{10}\) results in \(x = 0\).
Isolating the variable is a crucial skill in solving equations because it leads directly to the solution.
Checking Solutions
Once a potential solution is found, it is essential to verify that it satisfies the original equation. This is known as checking the solution. It ensures that there were no mistakes made during the equation-solving process.
To check, substitute the solution back into the original equation:\(7x + 3x = 0\).
  • Replace \(x\) with the found solution; in this case, \(x = 0\).
  • Calculate the left side of the equation to ensure it equals the right side.

Substituting gives \(7(0) + 3(0)\), which simplifies to \(0 + 0 = 0\).
Since both sides of the equation are equal, the solution \(x = 0\) is correct.
Checking your solutions helps confirm your results and ensures the accuracy of your work.

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Most popular questions from this chapter

For problems \(57-140\), solve each equation. $$ -4 k-6=7 $$

For problems \(57-140\), solve each equation. $$ \frac{a}{-5}-3=-2 $$

Translate each phrase or sentence to a mathematical expression or equation. Thirteen less than some number is equal to three times the number added to eight.

Find the value of \(3 \frac{1}{12}+4 \frac{1}{3}+1 \frac{1}{4}\).

For problems \(57-140\), solve each equation. $$ -2-\frac{2 x}{7}=3 $$
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