Problem 22 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$7(x-4)+3=5 x-9$$

Step-by-Step Solution

Answer

The solution for the equation is \(x = 8\).

Step 1: Distribute the 7

Start by distributing the 7 to both terms inside the parentheses on the left side of the equation:\[ 7(x-4) = 7x - 28 \]Thus, the equation becomes:\[ 7x - 28 + 3 = 5x - 9 \]

Step 2: Simplify the Equation

Combine any like terms on the left side of the equation:- Combine \(-28\) and \(+3\) to get \(-25\).Thus, the equation becomes:\[ 7x - 25 = 5x - 9 \]

Step 3: Move the Variables to One Side

Subtract \(5x\) from both sides to get all the variable terms on one side:\[ 7x - 5x - 25 = -9 \]This simplifies to:\[ 2x - 25 = -9 \]

Step 4: Move the Constants to the Other Side

Add 25 to both sides to isolate the term with \(x\):\[ 2x - 25 + 25 = -9 + 25 \]This simplifies to:\[ 2x = 16 \]

Step 5: Solve for x

Divide both sides by 2 to solve for \(x\):\[ \frac{2x}{2} = \frac{16}{2} \]This results in:\[ x = 8 \]

Key Concepts

Distributive Property

One of the fundamental tools in solving linear equations like the one in our problem is the distributive property. This property allows us to multiply a single term across terms inside parentheses. This process often simplifies the equation and makes it easier to handle.
For example, in our original equation, we have the expression \(7(x - 4)\). Using the distributive property, we multiply 7 by each term inside the parentheses:

• First, multiply 7 by \(x\), resulting in \(7x\).
• Next, multiply 7 by -4, resulting in \(-28\).

Thus, \(7(x-4)\) transforms into \(7x - 28\). Using this step makes the equation less complex and sets the stage for the remaining operations needed to find the solution.

Combining Like Terms

After distributing, the next crucial step involves combining like terms. This means adding or subtracting terms that have the same variable component. It helps in further simplifying the equation.
In our ongoing example, after distributing, the expression on the left side is \(7x - 28 + 3\). To simplify:

• Identify like terms: \(-28\) and \(+3\).
• These are both constant numbers and can be combined into \(-25\).

As a result, the equation evolves into \(7x - 25 = 5x - 9\). Combining like terms plays a key role in cleaning up the equation, making it more straightforward to solve.

Isolating Variables

Once we've simplified the equation by combining like terms, our next goal is to isolate the variable, which means getting 'x' by itself on one side of the equation. This often involves moving all terms with the variable to one side and constants to the other.
In our example, this looks like:

• First, we subtract \(5x\) from both sides, yielding \(2x - 25 = -9\).
• The \(x\) terms are now isolated to one side.
• Next, to further isolate \(2x\), we add 25 to both sides of the equation, transforming it to \(2x = 16\).

Isolating the variable simplifies the solution process and allows us to solve for 'x' directly.

Pre-Algebra Concepts

Finally, solving linear equations requires a foundational understanding of pre-algebra concepts. These include the basic operations of addition, subtraction, multiplication, and division, which collectively enable us to manipulate equations correctly.
Once we have the equation in the form \(2x = 16\) from isolating the variable, we use division:

• Divide both sides by 2 to solve for \(x\).
• So, \(2x \/ 2 = 16 \/ 2\).
• This simplifies the equation to \(x = 8\).

Understanding these pre-algebra concepts allows students to methodically work through more complex problems and ensures that each arithmetic operation is correctly applied. Master these concepts, and solving equations will become much more manageable.