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Chapter 3: Problem 1

Solve the equations. $$ 0.5 x-3=11 $$

Short Answer

Expert verified
Answer: The solution to the equation is \(x=28\).

Step by step solution

01

Add 3 to both sides of the equation.

We want to get rid of the constant term -3, so we'll add 3 to both sides of the equation: $$ 0.5x-3+3 = 11+3 $$ This simplifies to: $$ 0.5x = 14 $$
02

Divide both sides by 0.5.

Now we want to isolate the x term, so we'll divide both sides of the equation by 0.5: $$ \frac{0.5x}{0.5}=\frac{14}{0.5} $$ This simplifies to: $$ x=28 $$ So, the solution to the equation \(0.5x-3=11\) is \(x=28\).

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Key Concepts

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

Solving Equations
Solving equations may sound complicated, but it's all about balancing. Imagine a scale, and you need to ensure that both sides are equal. An equation like \(0.5x - 3 = 11\) is essentially asking you to find what value of \(x\) keeps the scale balanced. By performing operations such as adding, subtracting, multiplying, or dividing both sides by the same number, you maintain this equality. This way, you can find the number that makes the equation true.
When attacking any equation, keep these steps in mind:
  • Identify what operations and numbers are acting on the variable.
  • Use inverse operations to gradually eliminate these numbers and isolate the variable.
  • Keep the equation balanced by performing each operation on both sides.
Working through equations step by step can make seemingly complex problems much more approachable.
Isolation of Variables
In solving equations, one of the primary goals is to isolate the variable. This means getting the variable by itself on one side of the equation. In the equation \(0.5x - 3 = 11\), the variable \(x\) is being multiplied by 0.5 and decreased by 3. Our job is to reverse these operations to find the value of \(x\).
Think of isolation like peeling an onion鈥攜ou're removing layers (or operations) to reveal the core (the variable):
  • First, we want to deal with addition or subtraction around the variable. Add 3 to both sides to counteract the \(-3\). This simplifies the equation to \(0.5x = 14\).
  • Next, we need to address multiplication or division involving the variable. In this equation, \(x\) is multiplied by 0.5. So, we divide both sides by 0.5 to isolate \(x\).
Now the variable \(x\) stands alone, and we can see its value clearly.
Algebraic Manipulation
Algebraic manipulation is the art of rearranging and simplifying equations using various mathematical operations. It allows you to unlock the solution to an equation by carefully altering its form. For the equation \(0.5x - 3 = 11\), we use algebraic manipulation to transform it into a format where the solution is apparent.
Here's how manipulation works for this equation:
  • Addition: Start by adding 3 to both sides to simplify \(0.5x - 3 + 3 = 11 + 3\), resulting in \(0.5x = 14\). We've effectively removed the -3 from around the variable.
  • Division: Next, divide both sides by 0.5. This operation neutralizes the multiplication involving \(x\), bringing us to \(x = 28\).
Each manipulation step honors the balance principle of equations, ensuring we don't alter the fundamental equality while navigating towards the solution.

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Most popular questions from this chapter

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