Problem 74 - Algebra A Combined Function

Question

Solve each equation. \(9 x-4=7(x-2)\)

Step-by-Step Solution

Answer

The solution for the equation is \(x = -5\).

Step 1: Expand the Equation

First, we expand the right side of the equation. The original equation is \(9x - 4 = 7(x - 2)\). Expanding the equation on the right gives us \(9x - 4 = 7x - 14\).

Step 2: Isolate the Variable Terms

Next, we need to get all the terms with the variable \(x\) on one side. Subtract \(7x\) from both sides to do this: \((9x - 7x) - 4 = -14\) which simplifies to \(2x - 4 = -14\).

Step 3: Eliminate Constant Terms

Now, we will move the constant term \(-4\) on the left side of the equation to the right side by adding 4 to both sides: \(2x - 4 + 4 = -14 + 4\), which simplifies to \(2x = -10\).

Step 4: Solve for \(x\)

Finally, we solve for \(x\) by dividing both sides by 2: \(2x / 2 = -10 / 2\), which simplifies to \(x = -5\).

Key Concepts

Isolate Variable Terms

In solving linear equations, isolating variable terms means getting all the terms with the variable, typically denoted as \(x\), on one side of the equation. This is done by performing operations like addition or subtraction on both sides of the equation.
For our example, after expanding, we have the equation \(9x - 4 = 7x - 14\). To isolate the \(x\) terms, we need only to subtract \(7x\) from both sides.
This operation ensures that terms containing \(x\) gather on one side, leading to a simpler equation to work with, such as \(2x - 4 = -14\).
By focusing the variable on one side, we simplify the equation considerably, which helps in the next steps.

Expand Equations

When facing equations where expressions involve terms within parentheses, expanding those terms is crucial. It sets the foundation for a clear and manageable equation.
In our problem, \(9x - 4 = 7(x - 2)\), the right side includes the expression \(7(x - 2)\). Expanding involves distributing the 7 across terms inside the parentheses, leading to \(7 \times x\) and \(7 \times -2\).
This process gives us \(9x - 4 = 7x - 14\). It simplifies the equation, allowing us to see all terms clearly without lingering parentheses, setting the stage to easily isolate variable terms.

Eliminate Constant Terms

Eliminating constant terms is about moving constants in the equation to one side, ideally opposite the variable terms.
After isolating the variable terms, we reached \(2x - 4 = -14\). Here, the constant \(-4\) is on the side with the variable terms.
To move it, we perform the inverse operation, which is adding 4 to both sides. This gives us \(2x = -10\).
Removing constants like this helps us focus solely on the variable part of the equation, enabling straightforward solutions for the variable \(x\).

• Move constants by addition or subtraction.
• Simplify the equation progressively.

Step-by-Step Solution

Working through a step-by-step solution ensures that every part of the equation is handled systematically.
It begins with expanding terms, then moves to isolate and eliminate any constants, and finally focuses on the variable solution. Let's revisit the problem:

• Start by expanding: \(9x - 4 = 7x - 14\).
• Isolate terms: Subtract \(7x\) to get \(2x - 4 = -14\).


• Eliminate constants: Add 4 to obtain \(2x = -10\).
• Divide by 2: Solve for \(x\) to find \(x = -5\).

Approaching solutions in this organized way not only aids in understanding each step but also ensures accuracy. This methodical approach is the backbone of solving linear equations effectively.