Problem 33 - Elementary Algebra

Question

Solve. $$ 7 x-2+3 x=4+2 x-2 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{2}\).

Step 1: Combine Like Terms on Left Side

Start by combining the like terms on the left side of the equation. The terms are \(7x\) and \(3x\). Adding these gives \(10x\), so the equation becomes \(10x - 2 = 4 + 2x - 2\).

Step 2: Combine Like Terms on Right Side

On the right side, combine the numbers \(4\) and \(-2\). Adding them results in \(2\), so the equation simplifies to \(10x - 2 = 2 + 2x\).

Step 3: Move Variables to One Side

To get all the \(x\) terms on one side, subtract \(2x\) from both sides. The equation becomes \(10x - 2x - 2 = 2\), which simplifies to \(8x - 2 = 2\).

Step 4: Isolate the Variable

Add \(2\) to both sides to get \(8x = 4\).

Step 5: Solve for \(x\)

Divide both sides by \(8\) to find \(x\). This gives \(x = \frac{4}{8} = \frac{1}{2}\).

Key Concepts

Combining Like Terms

When solving algebra equations, combining like terms is an essential skill. Like terms in algebraic expressions are terms that have the same variables raised to the same power. This means you can efficiently add or subtract them.
For example, in the equation provided:

• Terms like \(7x\) and \(3x\) can be combined because they both have the variable \(x\). Similarly, on the right side, numerical constants like \(4\) and \(-2\) can be simplified since they are regular numbers.

Combining like terms is crucial because it makes your equation easier to solve. Think of it as simplifying the expression so that it becomes more manageable and less cluttered. Once you've combined all like terms, you'll have a clearer view of what steps to take next to solve for your variable. This step is like organizing a messy room—once everything is in its place, it's much easier to find what you need.

Isolating the Variable

The goal of solving an algebraic equation is to find the value of the variable. To do this, you must isolate the variable, meaning you arrange your equation so that the variable is by itself on one side of the equation. In our example, this was accomplished in a few steps.

• You start by moving any terms that include the variable to one side of the equation. In the solution, \(2x\) was subtracted from both sides.
• Once the variable terms were all on one side, the equation was further simplified by adding \(2\) to both sides so that all numbers (or constants) were on the opposite side.

Finally, once the coefficient in front of the variable was isolated together with the variable, we divided both sides by this coefficient (which was \(8\) in the example). By isolating the variable, you identify what the variable truly represents by itself, clearing the path to the solution.

Linear Equations

Linear equations are a type of equation that form a straight line when graphed. In these equations, the highest power of the variable is one, making them straightforward to solve.
In our example with the equation \(7x - 2 + 3x = 4 + 2x - 2\), you'll note that each term with \(x\) has an exponent of one. This consistent linearity is what allows us to use standard algebraic operations, such as combining like terms and isolating the variable, to solve for \(x\).

• Linear equations can usually be solved in several straightforward steps: combine like terms, get all variable terms on one side, isolate the variable, and then solve.
• They are different from quadratic equations, where the variable is squared, or other higher-degree polynomial equations.

By mastering these types of equations, you build a strong foundation in algebra that will support you in solving more complex problems in the future. Linear equations are often used to model real-world scenarios where relationships between two quantities are constant.