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The Distribution Property is a key concept to grasp when solving equations, especially when dealing with expressions in parentheses. It’s a mathematical rule that allows us to remove the parentheses by distributing, or multiplying, a number across the terms inside the parentheses. This simplification step is crucial as it prepares the equation for further solving steps.

When you see an expression such as \(a(b + c)\), the Distribution Property tells us that we can distribute \(a\) to both \(b\) and \(c\). This results in \(ab + ac\). For example, in the given equation \(0.9(6.2x - 5.9)\), applying the property means we multiply 0.9 with both 6.2x and -5.9.

• \(0.9 \times 6.2x = 5.58x\)
• \(0.9 \times -5.9 = -5.31\)

On the right side, a similar distribution takes place where 3.4 is multiplied across \(3.7x + 4.3\). This results in:

• \(3.4 \times 3.7x = 12.58x\)
• \(3.4 \times 4.3 = 14.62\)

By applying the Distribution Property, the expressions become simpler and ready for us to solve.

Linear equations are equations that involve variables raised only to the power of one. They are called 'linear' because when graphed, they create straight lines. Understanding how to recognize and solve these types of equations is fundamental to algebra.

The equation from the exercise \(0.9(6.2 x-5.9)=3.4(3.7 x+4.3)-1.8\) is a linear equation. Once you distribute the coefficients and simplify, the linear equation takes a clearer form. Typically, a linear equation might look like \(ax + b = cx + d\), where we can merge and compare terms to find the variable x.

• Identify and consolidate like terms on both sides of the equation.
• Rearrange terms to isolate the variable on one side of the equation.
• Solve for the variable, keeping the equation balanced by performing the same operation on both sides.

Solving this involves careful algebraic manipulation and ensures that we correctly find the value of x.

Algebraic manipulation refers to the process of rearranging and simplifying equations to isolate variables and solve for unknowns. This involves various techniques like combining like terms, moving terms across the equation, and division or multiplication to isolate the variable.

In our example equation, once we've applied the Distribution Property, we have expressions like \(5.58x - 5.31\) and \(12.58x + 14.62\) to manage. The goal is to get all terms with the variable, \(x\), on one side and constant terms on the other.

**Steps of Basic Algebraic Manipulation:**

• Subtract or add terms from both sides of the equation to align terms accordingly.
• If needed, divide or multiply each part of the equation to simplify further.
• Check your solution by substituting back into the original equation.

Through algebraic manipulation, we execute step-by-step operations to ensure all parts of the equation are balanced until we isolate the variable, finding our solution.