Problem 14 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ 4 x+7=9 x-13 $$

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Identify the Equation Type

This is a linear equation of the form \(ax + b = cx + d\), which means it can be solved for \(x\) by performing algebraic operations to isolate \(x\) on one side of the equation.

Step 2: Rearrange the Equation

First, subtract \(9x\) from both sides to get all \(x\)-terms on one side of the equation: \(4x - 9x + 7 = -13\).

Step 3: Simplify the Equation

Combine the \(x\)-terms: \(-5x + 7 = -13\).

Step 4: Isolate the Variable Term

Subtract 7 from both sides of the equation to move the constant term: \(-5x = -13 - 7\).

Step 5: Simplify the Right Side

Calculate \(-13 - 7\), resulting in \(-5x = -20\).

Step 6: Solve for x

Divide both sides by \(-5\) to isolate \(x\): \(x = \frac{-20}{-5}\).

Step 7: Simplify the Solution

Simplify \(\frac{-20}{-5}\) to get \(x = 4\).

Key Concepts

Solving Equations

When tackling a linear equation such as \(4x + 7 = 9x - 13\), the goal is to find the value of \(x\) that satisfies the equation. Solving equations involves a series of orderly steps, aimed at maintaining the balance between both sides of the equation.
For the equation \(ax + b = cx + d\), the first step typically involves identifying all the terms containing \(x\) and moving them to one side of the equation. This requires performing operations such as addition, subtraction, multiplication, or division on both sides.
The equation remains balanced as long as any operation done on one side is also applied to the other. In our given equation, this was carried out by subtracting \(9x\) from both sides. This approach sets the groundwork for effectively isolating the variable, which we will explore next.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to solve for a variable. This skill is essential for effectively solving linear equations. In our example, \(4x + 7 = 9x - 13\), algebraic manipulation is employed to group like terms and simplify the equation.
The objective is to simplify the equation while ensuring that each step brings us closer to isolating the variable of interest.

• First, recognize which terms should be moved. In this instance, subtract \(9x\) from both sides to make it \(-5x + 7 = -13\).
• Subsequently, combine the \(x\) terms to simplify. The equation now reads as \(-5x + 7 = -13\).

Additional operations may include factoring, distributing, or combining like terms. Algebraic manipulation provides a flexible toolbox for restructuring equations into a more solvable form.

Isolating Variables

Isolating variables is the technique of rearranging an equation to get \(x\) (or the variable of interest) by itself on one side of the equation. This is crucial for uncovering the value of \(x\) that solves the equation.
Starting from \(-5x + 7 = -13\), you need to eliminate constants from the \(x\)'s side by performing inverse operations. Subtracting 7 from both sides, we achieve \(-5x = -20\).
Next, to solve for \(x\), divide every term by the coefficient of \(x\), which is \(-5\) in this case. This means \(x = \frac{-20}{-5}\).

• Inverse operations undo the existing operation, like subtraction or division, to separate the variable.
• In our example, dividing by \(-5\) gives us \(x = 4\).

With \(x\) isolated, the solution can be directly identified. Effective isolation simplifies even complex equations, making them more approachable.