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Chapter 3: Problem 3

solve the equation. Show how to check your solution. $$4 x+3=11$$

Short Answer

Expert verified
The solution to the equation is x = 2

Step by step solution

01

Transfer the constant term

The first step is to get rid of the '+3' on the left side. To that end, subtract 3 from both sides: \(4x + 3 - 3 = 11 - 3\)This simplifies to: \(4x = 8\)
02

Solve for x

Then divide both sides of the equation by 4 to solve for x: \(4x/4 = 8/4\)Simplified, this gives: \(x = 2\)
03

Validate the Solution

Substitute the found value for x into the original equation to verify if it's indeed the solution:\(4*2 + 3 = 11\)This simplifies to 11=11, confirming the solution is correct

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Key Concepts

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

Solving Equations
Solving an equation means finding the value of the variable that makes the equation true. In our example, the equation is \(4x + 3 = 11\). The goal is to isolate \(x\) on one side of the equation. We begin by eliminating the constant on the side with the variable.

Here, we subtract 3 from both sides in the first step:
  • Starting with \(4x + 3 = 11\).
  • Subtract 3: \(4x + 3 - 3 = 11 - 3\).
  • This simplifies to \(4x = 8\).
Next, divide each side by 4 to solve for \(x\). This gives:
  • \(4x/4 = 8/4\).
  • Which simplifies to \(x = 2\).
Now we have solved the equation and found the value of \(x\). Solving equations often involves these steps: rearranging to isolate the variable and performing operations like addition, subtraction, multiplication, or division.
Algebra 1
Algebra 1 is foundational in learning about equations and their solutions. It involves understanding how to manipulate equations to find unknown values. An equation consists of two expressions set equal to each other, like \(4x + 3 = 11\).

In Algebra 1, some key tasks include:
  • Identifying and understanding variables: These stand for unknown numbers, like \(x\) in our equation.
  • Learning operations: Addition, subtraction, multiplication, and division are essential for solving equations.
  • Using the properties of equality: You can add, subtract, multiply, or divide both sides of the equation by the same number to maintain the equality, crucial for solving for variables.
Learning Algebra 1 enables students to solve real-world problems by creating and solving mathematical models. It's about finding the relationships between numbers and how altering one number or variable can affect others.
Equation Validation
Validating an equation means checking that the solution you found is correct. This is done by substituting the solution back into the original equation to see if both sides equalize.

For the equation \(4x + 3 = 11\), we found \(x = 2\) as the solution. To validate:\ul>
  • Substitute \(x = 2\) back: \(4(2) + 3 = 11\).
  • Calculate: \(8 + 3 = 11\).
  • This simplifies to \(11 = 11\).
    • Since both sides are equal, the solution is confirmed. Equation validation is a crucial step. It ensures the calculations were correct and the solution works within the context of the original equation. This process builds confidence in solving future math problems, reinforcing the understanding of algebraic principles.

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    Most popular questions from this chapter

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