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Chapter 1: Problem 24

Solve the given equation for \(x\). $$ x-20=8 $$

Short Answer

Expert verified
The solution is \( x = 28 \).

Step by step solution

01

Understand the Equation

The given equation is \( x - 20 = 8 \). This is a simple linear equation where we need to solve for \( x \), the unknown variable.
02

Isolate the Variable

To find the value of \( x \), we need to isolate it on one side of the equation. This can be done by eliminating the \(-20\) from the left side. We achieve this by adding \(20\) to both sides of the equation: \[ x - 20 + 20 = 8 + 20 \]
03

Simplify the Equation

Simplify both sides of the equation by carrying out the addition:\[ x = 28 \] Now, \( x \) is isolated, and we have determined its value.

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Key Concepts

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

Solving Linear Equations
When faced with a problem like "Solve the equation \(x - 20 = 8\)," our ultimate goal is finding the value of the unknown variable \(x\). Linear equations have the general form \(ax + b = c\), where \(x\) is the unknown we want to determine.
Solving an equation involves discovering which value for \(x\) makes both sides of the equation equal. Think of an equation as a balance scale: whatever you do to one side must be done to the other to maintain equality.
Here鈥檚 how to effectively solve linear equations:
  • Identify the equation and what you need to solve for.
  • Use inverse operations to move terms around.
  • Check your solution by plugging it back into the original equation.
By understanding these steps, solving equations becomes a logical process, aligning each step toward uncovering the unknown variable.
Isolation of the Variable
Isolating the variable is key when solving equations. In our example, we faced the equation \(x - 20 = 8\). To solve it, we need \(x\) on one side by itself. Here鈥檚 how you isolate \(x\):
  • Undo operations next to \(x\) using inverse operations. This means doing the opposite of what鈥檚 being done to \(x\). If there's subtraction, you add; if there's multiplication, you divide.
  • Whatever operation you perform, apply it equally to both sides of the equation to maintain balance.
For the equation \(x - 20 = 8\), adding 20 to both sides removes the "-20," leaving \(x\) isolated:
\[ x - 20 + 20 = 8 + 20 \]
This operation lets us solve for \(x\), leading us one step closer to the solution.
Simplification in Algebra
Simplification is about making the equation easier to interpret, leading us to the solution. Once you've isolated \(x\), carry out simplification to make the process clearer and direct.
In our example with the final equation \(x = 28\), it illustrates:
  • Complete any arithmetic operations to simplify each side as much as possible. Here, simple addition: \(8 + 20 = 28\).
  • Ensure the equation remains balanced and correctly expresses the relationship when the variable is isolated.
This simplicity ensures you're left with the clearest form of the solution. Associating the number 28 directly with the variable \(x\) completes the process, guaranteeing an accurate and simple answer to the initial equation.

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