The Aim When Solving an Equation

Introduction

In the previous section, we discussed the concept of simplifying expressions. Now, let’s shift our focus to solving equations. Equations are mathematical expressions that consist of two expressions set equal to each other using an equal sign (=). While the goal of simplifying expressions is to perform all possible operations and simplify the expression as much as possible, the objective of solving equations is to determine the value of the variable by isolating it on one side of the equation and a number on the other side.

The Process of Solving Equations

To achieve our goal of solving equations, we will follow two important steps:

1. Simplify each expression on both sides of the equal sign.
2. Use inverse operations to cancel out any unnecessary terms.

Example:

Let’s illustrate these steps with an example equation:

``5x - 4x - 6 = 18``

To start, we will simplify each side of the equation. Since there are no parentheses, exponents, or multiplication/division operations, we can proceed with addition and subtraction. The first step is to simplify `5x - 4x`, which results in `x`.

Cancelling Out with Inverse Operations

After simplification, we have the equation: `x - 6 = 18`. However, our goal is to isolate `x` and not the expression `x - 6`. To achieve this, we need to move the `-6` to the other side of the equation using the inverse operation. The inverse of subtracting `-6` is adding `6`. Therefore, we can add `6` to both sides of the equation.

``x - 6 = 18 + 6 + 6``

On the left side, `-6 + 6` cancels out to `0`, leaving us with `x`. On the right side, `18 + 6` results in `24`. Thus, our equation simplifies to `x = 24`. Notice how we simplified the equation by using the inverse of the operation we wanted to eliminate.

Understanding Cancelling Out

Cancelling out parts of an equation allows us to remove unwanted terms. However, there are important rules to follow:

1. Both sides of an equation must always be equal. Any operation performed on one side must also be performed on the other. For example, when we added `6` to the `-6` on the left side, we had to add it to the `18` on the right side.
2. To cancel out a term, we need to use its opposite or inverse operation. For instance, when we added `6` instead of subtracting it, we used the opposite of subtraction, which is addition.

Multi-step Equations

Now, let’s explore a more complex example:

``4(2x + 3) = 68``

First, we need to determine if any simplification is possible. According to our previous lesson on reading algebraic expressions, the number outside the parentheses signifies multiplication. Therefore, we can multiply `4` by `2x` and `4` by `3`, resulting in `8x` and `12`, respectively.

``8x + 12 = 68``

After simplification, we have `8x + 12 = 68`. To isolate `x`, we need to move both the `8` and `12` terms to the other side of the equation. Due to inverse operations, we will use addition/subtraction before multiplication/division. We’ll start by subtracting `12` from both sides:

``8x + 12 = 68 - 12 - 12``

Simplifying further, we find that `12 - 12` is `0`, leaving `8x` on the left side. On the right side, `68 - 12` is `56`. Now, we’ll divide both sides by `8`:

``8x = 56 / 8 / 8``

Dividing `56` by `8`, we get `7`. Therefore, `x = 7`. We have successfully solved the equation, and `x` is equal to `7`.

Practice!

Now, it’s time for you to practice solving equations on your own. Take a moment to simplify these expressions using the order of operations and cancelling out:

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Problem 1

Simplify this expression to find the value of `x`:

``6x + 23 = 74``

Answer: `x = 25`

Problem 2

Find the value of `y`:

``4(3y - 8) = 4``

Answer: `y = 4`

Problem 3

Find the value of `r`:

``300r - 60r + 102 = -380``

Answer: `r = -2`

Remember, simplifying equations requires attention to detail and following the correct sequence of steps. Double-check your answers, and don’t hesitate to check your work using the original equation and your derived value for the variable.

Continue practicing and mastering these skills, and you’ll become a pro at solving equations in no time!

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