Problem 18 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ 4 x-4=10 x-7 x $$

Step-by-Step Solution

Answer

The solution is x = 4.

Step 1: Simplify Each Side

Start by simplifying both sides of the equation if possible. The left side is already simplified: \(4x - 4\). On the right side, simplify by combining like terms: \(10x - 7x = 3x\). Thus, the equation becomes: \[ 4x - 4 = 3x \]

Step 2: Isolate the Variable

To solve for \(x\), get all terms involving \(x\) on one side of the equation. Subtract \(3x\) from both sides to isolate \(x\) on one side:\[ 4x - 3x - 4 = 0 \]This simplifies to:\[ x - 4 = 0 \]

Step 3: Solve for x

Now, solve for \(x\) by isolating it completely. Add 4 to both sides:\[ x - 4 + 4 = 4 \]Thus, \(x = 4\).

Step 4: Check the Solution

Substitute \(x = 4\) back into the original equation to verify the solution:Original equation: \(4x - 4 = 10x - 7x\)Substitute \(x = 4\): \(4(4) - 4 = 10(4) - 7(4)\)Calculate both sides: \(16 - 4 = 40 - 28\)This simplifies to \(12 = 12\).Since both sides are equal, \(x = 4\) is the correct solution.

Key Concepts

Simplifying Expressions

Simplifying an expression is like tidying up your room. The goal is to make things as straightforward as possible before attempting to solve a problem.

In our equation, the first step is to simplify both sides.

• The left side of the equation, \(4x - 4\), is already neat and needs no further adjustment.
• The right side has terms that can be combined. We go from \(10x - 7x\) to \(3x\).

This process involves reducing each side of the equation to its simplest form. By doing so, we make it easier to identify and work with the important parts of the equation.

Simple expressions mean fewer chances of making mistakes later on. This encourages a smoother solving process.

Combining Like Terms

Combining like terms is a crucial step in solving equations. It helps to manage components that are alike to simplify our work.

• Like terms are terms that have the same variable raised to the same power.
• In our example, on the right side of the equation \(10x - 7x\), both terms contain the variable \(x\).
• When we subtract \(7x\) from \(10x\), we combine them into a single term: \(3x\).

By managing like terms efficiently, we transform the equation into something more manageable. Remember, fewer terms mean fewer steps towards the solution, making the entire equation easier to handle.

Always ensure that terms are actually like terms before combining them. This will prevent you from making mistakes!

Isolating Variables

Isolating the variable is the strategy of getting the variable you are solving for on one side of the equation all by itself. The purpose of this is to figure out what \(x\) equals.

• In the simplified equation \(4x - 4 = 3x\), we need to take steps to have \(x\) by itself.
• We accomplish this by keeping \(x\) on one side and moving everything else to the other side.
• Subtracting \(3x\) from both sides gives us \(x - 4 = 0\).
• Finally, adding 4 to both sides isolates \(x\), resulting in \(x = 4\).

This step transforms our equation into something straightforward: \(x = 4\).

The biggest tip is to perform operations equally on both sides. This keeps the balance of the equation, which is essential in isolating the variable correctly.

Checking Solutions

The last step in solving any equation is checking the solution. This ensures that what you calculated is actually correct. After finding \(x = 4\), it’s good practice to verify by substituting it back into the original equation.

• The original equation is \(4x - 4 = 10x - 7x\).
• Substitute \(x = 4\) into the equation: \(4(4) - 4 = 10(4) - 7(4)\).
• Calculate both sides: \(16 - 4 = 40 - 28\).
• This simplifies to \(12 = 12\).

Seeing both sides equal confirms that our solution to the equation is accurate.

This is a helpful habit, ensuring confidence that you've successfully completed the problem. Always check your work to catch any potentially unnoticed errors.