**Solving linear equations** with **variables on both sides** is a fundamental skill in **algebra** that I consider essential for understanding a wide **range** of mathematical and **real-world problems.**

An **equation** of this sort typically looks like **$ax + b = cx + d$,** where **$x$ i**s the **variable** you’re looking to solve for, and** $a$, $b$, $c$**, and **$d$** represent constants.

My general strategy for tackling these **equations** involves a series of steps to isolate the **variable** on one side of the **equation.**

This means I’ll be **adding, subtracting, multiplying,** or dividing **throughout** to get **$x$** by itself, which simplifies the equation down to something like **$x = e$**, with **$e$** being the **solution.**

In dealing with these types of **equations,** it’s important to maintain the balance of the **equation**, as whatever I do to one side, I must also do to the other to **preserve equality.**

Sometimes, this involves **combining** like terms or using the **distributive property** to simplify each side of the **equation**; it’s like a mathematical dance where precision and **balance** are **key.**

Remember to check your solution by **substituting** it back into the **original equation** to ensure both sides equal out. Stay tuned as I walk you through this interesting and **engaging process,** sparking a light of understanding in the elegant dance of **algebra.**

## Steps for Solving Equations With Variables on Both Sides

When I encounter an **algebraic equation** with **variables** on both sides, my first step is to **simplify** each side of the equation if needed. This involves expanding expressions using the **distributive property**, combining **like terms**, and organizing the equation so it becomes easier to work with.

Here’s a breakdown in a table format:

Step | Action |
---|---|

1. Distribute | Apply the distributive property to remove parentheses. |

2. Combine like terms on each side | Add or subtract like terms on both sides of the equation. |

3. Get all variables on one side | Use addition or subtraction to move variables to one side. |

4. Get all constants on the opposite side | Move constants to the opposite side using opposite operations. |

5. Simplify the equation | Ensure each side of the equation is as simplified as possible. |

6. Solve for the variable | Divide or multiply to solve for the variable. |

Let’s consider the equation, $7y = 11x + 4x – 8$. I would combine the $x$ terms on the right first, getting $7y = 15x – 8$. Now, if I need $y$ on one side and $x$ on the other, I might decide to move the $x$ terms to the other side by subtracting $15x$ from both sides, resulting in $7y – 15x = -8$.

After that, if there are any **coefficients** attached to the **variable** I’m solving for, I’ll divide both sides by that coefficient. If I am solving for $y$, and the equation is $7y – 15x = -8$, I’d divide everything by 7 to isolate $y$:

$$ \frac{7y}{7} – \frac{15x}{7} = \frac{-8}{7} $$ $$ y – \frac{15x}{7} = \frac{-8}{7} $$

Now, if I need to solve for $y$ explicitly, I might rearrange the terms to show $y$ as a function of $x$, which would give me $y = \frac{15x}{7} – \frac{8}{7}$ as the final step.

This straightforward approach ensures the equation is balanced and leaves me with a clear solution for the **variable** in question.

## Solving Example Equation

When **I solve equations** with **variables on both sides**, I start by simplifying both sides separately. Let’s take the example of the **linear equation** $2x + 3 = x – 5$. My goal is to isolate the **variable**, x, on one side to find its value.

First, I use addition and **subtraction properties of equality** to get all the x terms on one side and the constants on the other. Subtracting x from both sides gives me $2x – x + 3 = x – x – 5$, which simplifies to $x + 3 = -5$.

Next, I apply **subtraction** to remove 3 from both sides, resulting in $x + 3 – 3 = -5 – 3$. Simplifying this, I get $x = -8$.

Here’s a breakdown of the steps I took in a more visual format:

Action | Equation | Purpose |
---|---|---|

Start with the original equation | $2x + 3 = x – 5$ | To begin simplification |

Subtract x from both sides | $x + 3 = -5$ | To get terms with x on one side |

Subtract 3 from both sides | $x = -8$ | To isolate the variable |

After simplifying, I realize that the equation has led me to a **true statement**: $x = -8$ is the solution indicating the value of the variable that satisfies the **equation**.

It’s also important to **combine like terms** and utilize the **distributive property** appropriately if the **equation** were to have more complex expressions.

By performing the correct **multiplication** or **division** based on the **properties of equality**, I ensure the equation remains balanced and arrive at an accurate solution.

## Conclusion

Solving **linear equations** with **variables** on both sides has been a focus throughout this **discussion.**

I’ve **demonstrated** how to bring the **variables** to one side and the constants to the other, which often involves the use of the **addition** or **subtraction** property of equality. Remember to always perform the same **operation** on both sides to maintain the balance of the **equation.**

Once the **variables** are **isolated,** you might need to use the division property to find the **variable’s** value. The key to **solving** these **equations** lies in following a **systematic approach: distribute, combine** like terms, and **isolate** the **variable**.

For example, when presented with an equation like **$3x + 7 = 2x – 5$**, my first step is to eliminate the **$x$** from one side by subtracting **$2x$** from both sides, simplifying it to **$x + 7 = -5$**. Then, I isolate $x$ by subtracting $7$ from both sides, resulting in **$x = -12$**.

By taking these steps, I ensure that each **equation** is solved accurately, and the true value of the **variable** is revealed.

Finally, checking my work by **substituting** the **solution** back into the original equation **guarantees** the **integrity** of the results.

This **methodical** approach not only makes complex problems more **manageable** but also instills confidence in my ability to **tackle** similar **challenges** in the future.