Problem 28 - Algebra for College Students

Question

Solve each equation. \(x-4-4 x=6 x+9-8 x\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -13\).

Step 1: Combine Like Terms on Both Sides

Simplify both sides of the equation by combining like terms. For the left side, combine the terms involving \(x\): \(x - 4x\) becomes \(-3x\). The left side simplifies to \(-3x - 4\). For the right side, combine \(6x\) and \(-8x\), which simplifies to \(-2x\). The right side becomes \(-2x + 9\). Hence the equation becomes \(-3x - 4 = -2x + 9\).

Step 2: Isolate the Variable Term

Move the \(-2x\) term from the right to the left side of the equation. This can be done by adding \(2x\) to both sides: \(-3x + 2x - 4 = -2x + 2x + 9\), which simplifies to \(-x - 4 = 9\).

Step 3: Solve for the Variable

Add 4 to both sides to isolate \(-x\): \(-x - 4 + 4 = 9 + 4\). This becomes \(-x = 13\).

Step 4: Solve for the Positive Variable

Finally, multiply both sides by \(-1\) to solve for \(x\): \(-1\times (-x) = -1 \times 13\), resulting in \(x = -13\).

Key Concepts

Combining Like Terms

When working with linear equations, one of the first steps is often combining like terms. Like terms are terms in an equation that have the same variables raised to the same power. For example, in the equation provided, you have several terms with the variable \(x\): \(x\), \(-4x\), \(6x\), and \(-8x\). These are all like terms because they involve the variable \(x\) raised to the first power.

Combining these terms means performing the arithmetic operation (addition or subtraction) among them. On the left-hand side of the equation, combining like terms \(x - 4x\) results in \(-3x\). On the right-hand side, \(6x - 8x\) simplifies to \(-2x\).

By combining like terms, the original equation \(x - 4 - 4x = 6x + 9 - 8x\) simplifies to \(-3x - 4 = -2x + 9\), setting the stage for the next steps in solving the equation.

Isolating the Variable

After combining like terms, the next strategy is to isolate the variable. This process helps simplify the equation, making it easier to solve for the variable. With the equation \(-3x - 4 = -2x + 9\), we aim to get all the terms involving \(x\) on one side.

To achieve this, you can either move the terms from the right to the left or vice versa, depending on personal preference or which seems simpler. In this case, adding \(2x\) to both sides effectively moves the \(-2x\) from the right to the left, resulting in a new equation:

\(-3x + 2x - 4 = -2x + 2x + 9\).

The equation simplifies to \(-x - 4 = 9\). This process ensures one side is free of the variable, allowing for easier solutions in subsequent steps.

Solving for the Variable

With the variable isolated to one side, you can solve for it by removing any constants that share the same side. In the simplified equation \(-x - 4 = 9\), adding 4 to both sides allows you to cancel out the \(-4\):

\(-x - 4 + 4 = 9 + 4\).

This operation results in \(-x = 13\). With only the variable \(-x\) left on one side, the equation becomes more straightforward. However, notice that the variable is negative, which brings us to the final adjustments needed.

Step-by-Step Algebra Solutions

Algebra often involves several steps to transition from a complex equation to an easily solvable form. In our problem, the final step after isolating \(-x\) involved solving for a positive variable. This requires converting the negative variable into its positive form by multiplying or dividing both sides by -1.

Applying this to \(-x = 13\), you perform the multiplication:

\(-1 \times (-x) = -1 \times 13\).

The equation resolves to \(x = -13\), giving you the solution to the original equation. Step-by-Step solutions such as these provide a clear path from problem to solution, demonstrating methodically how each move contributes to simplifying the equation, making complex algebra problems more approachable.