Problem 11 - Prealgebra

Question

Solve each equation. Check your solution. $$3 x+1=7$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\).

Step 1: Isolate the Variable

To solve the equation, we first need to isolate the variable \(x\) on one side. Begin by subtracting 1 from both sides of the equation:\[ 3x + 1 - 1 = 7 - 1 \]Simplifying both sides, we get:\[ 3x = 6 \]

Step 2: Solve for the Variable

Now, divide both sides of the equation by 3 to solve for \(x\):\[ \frac{3x}{3} = \frac{6}{3} \]This simplifies to:\[ x = 2 \]

Step 3: Check the Solution

Substitute \(x = 2\) back into the original equation to verify the solution:\[ 3(2) + 1 = 7 \]Perform the multiplication and addition:\[ 6 + 1 = 7 \]Since both sides of the equation are equal, \(x = 2\) is indeed the correct solution.

Key Concepts

Understanding Algebra Basics

Algebra is all about finding the unknown by understanding the relationship between numbers or variables. When solving linear equations, like the one in this exercise, we often encounter an equation that includes variables, constants, and coefficients. The very first step in learning algebra involves getting familiar with these parts. For instance, in the equation \(3x + 1 = 7\):

• Variable: \(x\) is the variable, which is the unknown value we aim to solve for.
• Coefficient: The number 3 in front of \(x\) is its coefficient, indicating that \(x\) is multiplied by 3.
• Constant: The numbers 1 and 7 are constants; they don’t change along with the variable.

The key goal in solving algebraic equations is to find the value of the variable that makes the equation true. Algebra provides a powerful way to discover these values in a systematic manner.

Mastering Equation Solving Steps

Solving linear equations involves a series of logical steps aimed at isolating the variable. Here’s a closer look at the steps used to solve the equation \(3x + 1 = 7\):

• Step 1 - Isolating the Variable: The primary aim here is to get the variable on one side of the equation. You do this by performing arithmetic operations. In our example, subtract 1 from both sides to neutralize the constant on the side with the variable. This is based on the principle of maintaining balance in equations.


• Step 2 - Solving for the Variable: Once the equation is \(3x = 6\), the target shifts to finding the exact value of \(x\). Since \(3x\) means 3 times \(x\), you do the opposite operation — division. By dividing both sides by 3, you isolate \(x\), thus arriving at its value \(x = 2\).

Following these equation solving steps helps in consistently reaching the correct solution, while practice helps in carrying out these efficiently.

Techniques for Checking Solutions

It's crucial to confirm that the value you've found for the variable truly satisfies the original equation. Checking your solution involves substituting back to see if the equation holds true. Here’s how this is done with our equation scenario:

• Substitution: Replace \(x\) in the original equation \(3x + 1 = 7\) with the found value \(2\). This becomes \(3(2) + 1 = 7\).


• Validation through Calculation: Perform the multiplication first, then the addition, resulting in \(6 + 1 = 7\). Both sides of the equation equaling 7 confirms \(x = 2\) is correct.

Employing these checks is essential, as they help confirm that the solution is accurate and gives you confidence moving forward in solving more complex equations.