Problem 13 - Algebra 1

Question

Solve the equation and describe each step you use. $$ -8 x+7=4 x-5 $$

Step-by-Step Solution

Answer

The solution to the equation \(-8x+7 = 4x-5\) is \(x = 1\).

Step 1: Rearrange the equation

To find the value of \(x\), you should begin by moving terms involving \(x\) to one side of the equation and the rest to the other side. You do this by adding \(8x\) to both sides and adding \(5\) to both sides of the equation: \(-8x + 8x + 7 + 5 = 4x + 8x - 5 + 5\). This simplifies to \(12 = 12x\)

Step 2: Isolate Variable x

Now get \(x\) on its own by dividing both sides of the equation by \(12\): \(\frac{12}{12} = \frac{12x}{12}\). This gives the result: \(x = 1\).

Key Concepts

Linear Equations and Their Structure

A linear equation is a type of equation where each term is either a constant or the product of a constant with a single variable. These equations are called linear because they graph as straight lines. Linear equations are foundational in algebra because they describe relationships where the change between variables is constant. In the equation \[ -8x + 7 = 4x - 5 \] you can see that every term fits the linear criteria. There are no exponents or powers attached to any of the variable terms, which makes this a single-variable linear equation. Understanding linear equations is important for solving more complex types of equations, and recognizing them is the first step to mastering algebra. The goal in solving these equations is often to find the variable value that makes the equation true. Solutions may require algebraic techniques like moving terms and simplifying expressions.

Algebraic Manipulation Techniques

Algebraic manipulation involves using various mathematical operations to rearrange equations into a more useful form. This allows us to find the value of unknown variables. In the equation \[-8x + 7 = 4x - 5\] you have to perform operations to simplify and uncover the variable, which here is \(x\). The first step in this manipulation process is to move terms involving the variable \(x\) to one side and constant terms to the other side:

• **Add** \(8x\) to both sides of the equation: Now, the left side becomes \(7\) and the right side becomes \(12x - 5\).
• **Add** \(5\) to both sides of the equation: By doing so, you attain \(12 = 12x\).

Each of these steps uses the basic principle of adding or subtracting the same quantity from both sides of an equation, preserving equality. This technique is crucial because it maintains the balance inherent to equations while bringing you closer to solving the unknown variable.

Variable Isolation through Division

Variable isolation is the process of moving all terms involving the variable of interest to one side of the equation, eventually leaving the variable alone. In our example \[ 12 = 12x \] the variable already stands alone on one side, but it is multiplied by a constant (12). To isolate \(x\), you simply need to divide both sides by 12.

• This gives: \( \frac{12}{12} = \frac{12x}{12} \)
• Thus, resulting in: \(1 = x \)

The action of division cancels out the multiplication of 12 with \(x\), effectively isolating it. Division is often used to eliminate coefficients attached to variables, making it a fundamental tool in solving equations. Mastering variable isolation through division is essential for solving not only linear equations but also more complicated algebraic equations.