Problem 56 - Prealgebra

Question

Solve the equation. \(-3.3(-6.3 x+4.2)-5.3=1.7(6.2 x+3.2)\)

Step-by-Step Solution

Answer

The solution is approximately 2.4.

Step 1: Distribute on the Left Side

To start, we distribute -3.3 across the terms inside the parenthesis on the left-hand side:\[-3.3(-6.3x + 4.2) = (-3.3) imes (-6.3x) + (-3.3) imes 4.2.\]Calculating these products:\[(-3.3) imes (-6.3x) = 20.79x,\]\[(-3.3) imes 4.2 = -13.86.\]Thus, the left side becomes:\[20.79x - 13.86 - 5.3.\]

Step 2: Simplify the Left Side

Next, we simplify the left side by combining constants \[-13.86 - 5.3.\]This gives us:\[20.79x - 19.16.\]

Step 3: Distribute on the Right Side

Distribute 1.7 across the terms inside the parenthesis on the right-hand side:\[1.7(6.2x + 3.2) = 1.7 imes 6.2x + 1.7 imes 3.2.\]Calculating these products:\[1.7 imes 6.2x = 10.54x,\]\[1.7 imes 3.2 = 5.44.\]So the right side becomes:\[10.54x + 5.44.\]

Step 4: Equate the Expressions and Simplify

Equate the expressions from both sides of the equation:\[20.79x - 19.16 = 10.54x + 5.44.\]

Step 5: Isolate x

First, move the terms involving x to one side and the constants to the other side:Subtract \(10.54x\) from both sides:\[20.79x - 10.54x = 5.44 + 19.16.\]This simplifies to:\[10.25x = 24.6.\]

Step 6: Solve for x

To find the value of \(x\), divide both sides by 10.25:\[x = \frac{24.6}{10.25}.\]Calculating this division results in:\[x \approx 2.4.\]

Key Concepts

Understanding the Distributive Property

The distributive property is a key concept in algebra that allows us to multiply a single term across terms within a parenthesis. When solving linear equations, this property becomes essential. It ensures each component inside the parenthesis is evenly distributed or multiplied by the number or variable outside the bracket.
In our problem, we see the expression \(-3.3(-6.3x + 4.2)\) on the left side of the equation. To distribute \(-3.3\), we multiply it with every term in the parenthesis:

• First, \((-3.3) \times (-6.3x) = 20.79x\).
• Then, multiply \((-3.3) \times 4.2 = -13.86\).

This results in a new expression on the left side without changing the equation's value, effectively simplifying it to \(20.79x - 13.86 - 5.3\). Promoting these steps is the initial ground for solving the equation successfully.
The same property is applied on the right side with \(1.7(6.2x + 3.2)\). This consistent use of distributive property helps unravel complex expressions into simpler equations where the actual value of the variable can be isolated later.

Combining Like Terms

Combining like terms is an integral part of simplifying expressions. It means to add or subtract terms in an algebraic expression that share the same variable raised to the same power.
From the equation after applying the distributive property on the left side, we get\(20.79x - 13.86 - 5.3\). Here,

• The constants, \(-13.86\) and \(-5.3\), can be combined by simple addition, resulting in \(-19.16\).

This combines the entire expression to \(20.79x - 19.16\), enabling a more manageable equation on the left side.
This process is crucial because having fewer terms makes it significantly easier to solve for the unknown variable. While this might seem a small step, combining like terms transforms lengthy expressions into simplistic equations, leading towards solving the variable efficiently.

Isolating the Variable

Isolating the variable is often the final goal in solving linear equations. It involves manipulating the equation so that the variable is by itself on one side of the equation, usually the left side.
In our equation, we reached a stage where\(20.79x - 19.16 = 10.54x + 5.44\). The first step towards isolating x is to move all terms involving \(x\) to one side and constants to the other. To achieve this:

• Subtract \(10.54x\) from both sides: \(20.79x - 10.54x = 5.44 + 19.16\), simplifying it to \(10.25x = 24.6\).

Now with \(10.25x = 24.6\), the variable is nearly isolated. All that's left is to get \(x\) alone:

• Simply divide both sides by \(10.25\): \[x = \frac{24.6}{10.25}.\]

Calculating this division gives \(x \approx 2.4\).
By neatly following these steps—applying operations that maintain overall equality—solving for \(x\) becomes efficiently streamlined. This method is key in unraveling linear equations, enabling you to solve variables swiftly and correctly.