Problem 10 - Elementary Algebra

Question

Solve. $$5 x+5=3 x+1$$

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Simplify the Equation

To start solving the equation, simplify both sides. The equation given is: \[5x + 5 = 3x + 1\] First, identify the terms that are similar, which are both expressions of \(x\). Your next move will simplify the equation by bringing like terms together.

Step 2: Move Variable Terms to One Side

Subtract \(3x\) from both sides of the equation to group all variable terms on one side. This gives: \[5x + 5 - 3x = 3x + 1 - 3x\] Simplifying this will result in: \[2x + 5 = 1\]

Step 3: Isolate the Variable Term

Subtract 5 from both sides to isolate the term with the variable. Thus, you have: \[2x + 5 - 5 = 1 - 5\] Which simplifies to: \[2x = -4\]

Step 4: Solve for the Variable

Divide both sides by 2 to solve for \(x\): \[\frac{2x}{2} = \frac{-4}{2}\] This simplifies to: \[x = -2\]

Step 5: Conclusion

We have solved for \(x\) and found that in the equation \(5x + 5 = 3x + 1\), the value of \(x\) is \(-2\).

Key Concepts

Solving Equations

Solving linear equations is an essential skill in algebra that involves finding the value of the variable that makes the equation true. In our exercise, we started with the equation \(5x + 5 = 3x + 1\). Solving it required simplifying and rearranging the terms.

Generally, the process includes several key steps:

• Simplifying the equation by combining like terms or applying basic arithmetic operations such as addition, subtraction, multiplication, or division to simplify each side of the equation.
• Rearranging the equation to bring all terms with the variable onto one side and constant terms onto the other. This helps streamline the problem to its simplest form.
• Applying operations on both sides of the equation evenly, ensuring that the equality remains balanced. This also involves isolating the term with the variable for a clear view of its value.

Solving equations is not about guesswork; it's a structured process that builds foundational thinking skills critical in mathematics and science.

Algebra Concepts

Algebra is the backbone of mathematics that deals with symbols and the rules for manipulating these symbols to solve problems. The central idea is to express real-life situations with mathematical expressions and equations.

In the exercise, we used basic algebra concepts to solve the equation \(5x + 5 = 3x + 1\). Key concepts include:

• Like Terms: Terms that have the same variable raised to the same power. For instance, \(5x\) and \(3x\) are like terms because they both contain the variable \(x\).
• Operations on Equations: In algebra, performing the same operation on both sides of an equation keeps the equation balanced, ensuring a valid solution.
• Expressions and Equations: Expression involves numerical values, operators, and sometimes variables, while equations are equal expressions denoting a balance.

Mastering these concepts forms the basis for solving more complex problems in mathematics.

Variable Isolation

The goal of variable isolation is to manipulate an equation until the variable of interest is alone on one side. This step is crucial to solving equations and is often done after simplifying and rearranging the terms.

In our equation, \(5x + 5 = 3x + 1\), variable isolation took a few strategic manipulations:

• First, we subtracted \(3x\) from both sides to start collecting all \(x\) terms on one side, resulting in \(2x + 5 = 1\).
• Next, we subtracted 5 from both sides to further isolate \(x\) terms, leading to \(2x = -4\).
• Finally, we divided both sides by 2 to solve for \(x\), achieving \(x = -2\).

By isolating the variable, you can clearly understand the value it takes to maintain the equation's balance. This concept is fundamental as equations become more complex in higher levels of math.