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Chapter 11: Problem 84

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

Short Answer

Expert verified
\( x = \frac{5}{3} \)

Step by step solution

01

Isolate the variable term

To isolate the variable term, subtract 2 from both sides of the equation: \[ 3x + 2 - 2 = 7 - 2 \]This simplifies to:\[ 3x = 5 \]
02

Solve for the variable

To solve for \(x\), divide both sides by 3:\[ \frac{3x}{3} = \frac{5}{3} \]This simplifies to:\[ x = \frac{5}{3} \]

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Key Concepts

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

Isolate the Variable
Solving linear equations often starts with the goal of getting the variable all by itself on one side of the equation. This process is known as isolating the variable. Let鈥檚 break down how we achieve this.

1. **Identify the variable**: In our initial equation, the variable we're interested in is \( x \).
  • First, look at your equation: \(3x + 2 = 7\).
  • The variable term here is \(3x\). Our task is to remove any numbers that are added or subtracted from this variable term.
2. **Eliminate constants**: After identifying the variable term, location of the variable, and the constant, calculate what needs to be moved.
  • In our case, subtract 2 from both sides:
  • Write it out as: \(3x + 2 - 2 = 7 - 2\).
  • You're left with \(3x = 5\). Now the variable \(x\) is nicely isolated with only the 3 next to it.
Divide Both Sides
After isolating the variable term, the next step is to "undo" whatever multiplication or division is left on the variable. This can often simply mean dividing both sides of the equation by the same number.Let's discuss how this works in our equation.

1. **Identify the coefficient**:
  • Our equation after isolation is \(3x = 5\).
  • Here, 3 is the coefficient of \(x\).
2. **Divide to solve**:
  • We need to remove the 3 next to \(x\). Thus, we divide both sides of the equation by 3.
  • Write: \(\frac{3x}{3} = \frac{5}{3}\).
  • After dividing, the equation simplifies to \(x = \frac{5}{3}\). This division step effectively frees \(x\) from its coefficient, giving us the solution.
Simplify the Equation
Simplifying an equation is a vital process in mathematics, where you rewrite it in its simplest, most efficient form. This makes equations easier to solve and understand.For our example, simplifying occurred at two main stages.

1. **After isolating**:
  • Initially, we had \(3x + 2 = 7\). After isolating, we approached \(3x = 5\).
  • This form is already a result of simplification, where we only deal with \(3x\) instead of the full original equation.
2. **Final simplification**:
  • Once we divide both sides by 3, the equation becomes \(x = \frac{5}{3}\).
  • This is the simplest form of the solution.
By simplifying, the equation is less cluttered, making it easier to see the solution clearly and confirm accuracy. This process ensures that each operation effectively simplifies the relationship between numbers and variables, leading to ease in interpretation and calculation.

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Most popular questions from this chapter

Translate each phrase or sentence into a mathematical expression or equation. A number divided by three, minus the same number multiplied by six, is one more than the number.

For problems \(17-46\), find the value of each expression. $$ -a+3 b-6, \text { if } a=-3 \text { and } b=0 $$

Translate each phrase or sentence to a mathematical expression or equation. When twelve is subtracted from five times some number, the result is two less than the original number.

Translate each phrase or sentence to a mathematical expression or equation. Ten times a quantity increased by two is nine.

Three consecutive even integers add to -36 . What are they?
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