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Chapter 3: Problem 42

Solve each equation. \(x-63=10 x\)

Short Answer

Expert verified
The solution is \(x = -7\).

Step by step solution

01

Set Up the Equation

The given equation is \(x - 63 = 10x\). The goal is to solve for \(x\).
02

Isolate \(x\) on One Side

Subtract \(x\) from both sides to begin isolating \(x\):\[x - 63 - x = 10x - x\]which simplifies to:\[-63 = 9x.\]
03

Solve for \(x\)

To solve for \(x\), divide both sides by 9 to isolate \(x\):\[x = \frac{-63}{9}.\]
04

Simplify the Result

Simplify \(\frac{-63}{9}\) by dividing the numerator and the denominator by their greatest common factor, which is 9:\[x = \frac{-63 \div 9}{9 \div 9} = \frac{-7}{1} = -7.\]

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Key Concepts

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

Isolation of Variable
In solving linear equations, one common goal is to isolate the variable. This means getting the variable, often represented as \(x\), alone on one side of the equation. In the equation \(x - 63 = 10x\), isolating the variable involves moving terms strategically from one side to the other.

To start, notice that \(x\) appears on both sides of the equation. By subtracting \(x\) from both sides, we are working to isolate \(x\) on one side. This simplifies as \(-63 = 9x\). Now, \(x\) is neatly isolated with its coefficient on one side of the equation, easing the path to finding its value.
  • Subtract terms involving \(x\) from both sides to bring them together.
  • The goal is to have the variable on one side with a constant on the other.
Simplifying Expressions
Simplifying expressions is a crucial step in solving equations. Once the variable is isolated, we often face expressions that are not in their simplest form. In our example, after isolation, the equation becomes \(-63 = 9x\). This expression is relatively straightforward, but further steps might involve more complex expressions.

Simplifying involves performing operations to reduce expressions to their most basic form. This might involve combining like terms, reducing fractions, or using common factors, as seen when simplifying \(\frac{-63}{9}\).
  • Combine like terms to reduce clutter.
  • Factor expressions to find the simplest form.
Solving for x
After isolating \(x\) and simplifying the equation, our final step is solving for the variable. Here, solving means determining the value of \(x\) that satisfies the equation. In \(-63 = 9x\), we need to solve for \(x\).

To do this, divide both sides by 9 to isolate \(x\). This process effectively turns the equation into \(x = \frac{-63}{9}\). Now, simplify the fraction: both 63 and 9 can be divided by their greatest common factor, 9. Thus, \(\frac{-63}{9} = -7\).
  • Divide both sides by the coefficient of \(x\) to solve for it.
  • Always simplify the result to find the neatest form of \(x\).

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Most popular questions from this chapter

Write each phrase as a variable expression. Use \(x\) to represent "a number." The product of -6 and the sum of a number and 15

A classmate tries to solve \(3 x=39\) by subtracting 3 from both sides of the equation. Will this step solve the equation for \(x\) ? Why or why not?

In retailing, the retail price \(P\) of an item can be computed using the equation \(P=C+M,\) where \(C\) is the wholesale cost of the item and \(M\) is the amount of markup. Slidell Feed and Seed sells a bag of cat food for \(\$ 12 .\) If the store paid \(\$ 7\) for the cat food, what is the markup on the cat food?

A classmate shows you steps for solving an equation. The solution does not check, but the classmate is unable to find the error. For each set of steps, check the solution, find the error, and correct it. $$\begin{aligned}37 x+1 &=9(4 x-7) \\\37 x+1 &=36 x-7 \\\37 x+1-1 &=36 x-7-1 \\\37 x &=36 x-8 \\\37 x-36 x &=36 x-8-36 x \\\x &=-8\end{aligned}$$

Solve each equation. \(23 x+8-25 x=7-9\)
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