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Chapter 5: Problem 5

Solve and check. $$3 x+1=10$$

Short Answer

Expert verified
The solution to the equation is \(x = 3\).

Step by step solution

01

Isolate the variable term

Start by isolating the \(x\) term by taking all other terms to the other side of the equation. You can do this by subtracting 1 from both sides of the equation: \(3x + 1 - 1 = 10 - 1\), which simplifies to \(3x = 9\).
02

Solving for \(x\)

Next, to solve for \(x\), divide both sides of the equation by 3, the coefficient of \(x\), to obtain \(x = 3\).
03

Checking the solution

To ensure that the solution is correct, insert the value of \(x\) into the initial equation: \(3*3 + 1\). If this equals 10, then 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 Linear Equations
When faced with solving linear equations, you're tasked with finding the unknown variable's value. Linear equations can be straightforward, consisting of just one variable. Let's break it down:
  • Perform operations: Use basic arithmetic operations such as addition, subtraction, multiplication, and division to isolate the variable.
  • Maintain equality: Whatever operation you perform on one side of the equation, remember to do the same on the other side to maintain balance.
In the example of the equation \(3x + 1 = 10\), we start by simplifying both sides to gradually solve for \(x\). With these strategies, solving any linear equation becomes manageable.
Isolation of Variables
The isolation of variables is crucial in solving equations. It refers to the process of manipulating the equation to get the unknown variable by itself on one side.
Here's how to approach it:
  • Eliminate constants: Start by moving all constants (numbers without variables) from the side containing the variable to the other side. In our example \(3x + 1 = 10\), subtract 1 from both sides to get \(3x = 9\).
  • Remove coefficients: After handling the constants, divide or multiply to eliminate any coefficients attached to the variable. Divide both sides by 3 to isolate \(x\), resulting in \(x = 3\).
Isolation simplifies the equation and directs us toward the solution. Remember, always ensure that each step clears the equation, getting you closer to solving for the variable.
Checking Solutions
Once you've solved a linear equation, checking your solution is essential. It confirms that the value you've found for the variable is correct, ensuring no mistakes were made during the process.
The check involves substituting the value of the variable back into the original equation to verify both sides equal. For the equation \(3x + 1 = 10\) and the solution \(x = 3\):
  • Substitute \(x = 3\) back into the equation: \(3(3) + 1 = 9 + 1 = 10\).
  • If the simplified result on one side matches the other side of the equation, your solution is confirmed as correct.
This step is critical and should never be skipped, as it validates your entire problem-solving process.

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