Problem 52 - Prealgebra

Question

Solve the equation. \(-1.1 x+1.3(x+1.3)=19.88\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 90.95\).

Step 1: Distribute the Multiplication

Expand the equation by distributing the term in the parentheses. Multiply 1.3 by both terms in the parentheses: \(1.3(x + 1.3) = 1.3x + 1.3 imes 1.3\).

Step 2: Simplify the Equation

Replace the expanded expression in the equation to form a new expression: \(-1.1x + 1.3x + 1.3 imes 1.3 = 19.88\). Since \(1.3 \times 1.3 = 1.69\), the equation becomes: \(-1.1x + 1.3x + 1.69 = 19.88\).

Step 3: Combine Like Terms

Combine the like terms of \(x\) in the equation: \((-1.1x + 1.3x) + 1.69 = 19.88\) simplifies to \(0.2x + 1.69 = 19.88\).

Step 4: Isolate the Variable Term

To isolate the term with \(x\), subtract 1.69 from both sides of the equation:\(0.2x = 19.88 - 1.69\). Calculate the right side: \(0.2x = 18.19\).

Step 5: Solve for the Variable

Finally, solve for \(x\) by dividing both sides by 0.2:\(x = \frac{18.19}{0.2}\). Perform the division to find \(x\): \(x = 90.95\).

Key Concepts

Distributive Property

The distributive property is a valuable tool when solving linear equations. It allows us to simplify expressions by distributing a single term over terms inside parentheses. In the given problem, we have the expression * \(1.3(x + 1.3)\)
Here, the number \(1.3\) multiplies each term inside the parenthesis. Breaking it down,

• First, we multiply \(1.3\) by \(x\), giving us \(1.3x\).


• Next, we multiply \(1.3\) by \(1.3\), resulting in \(1.3 \times 1.3 = 1.69\).

Once distributed, these products form a new expression: \(1.3x + 1.69\).
This process is crucial in managing equations with parenthetical terms, ensuring all elements are accounted for in the solution.

Combining Like Terms

Combining like terms simplifies an equation, making it easier to solve. In our linear equation, after using the distributive property, the expression was* \(-1.1x + 1.3x + 1.69 = 19.88\).
Here, we notice the terms \(-1.1x\) and \(1.3x\) both contain the variable \(x\). These are 'like terms' because they share the same variable component. To combine them,

• Add their coefficients, \(-1.1 + 1.3\), to calculate \(0.2x\).

This consolidation results in a more straightforward equation: * \(0.2x + 1.69 = 19.88\).
Efforts to combine like terms streamline the equation and help focus on solving the variable more efficiently, which is essential for moving forward in the problem.

Variable Isolation

Variable isolation is the art of positioning the variable on one side of the equation, allowing its value to be easily found. Once we've simplified the equation to* \(0.2x + 1.69 = 19.88\),
we strive to isolate \(x\). This means eliminating other numbers from its side using algebraic operations. Here, we start by:

• Subtracting \(1.69\) from both sides, obtaining \(0.2x = 19.88 - 1.69\).
• That simplifies to \(0.2x = 18.19\).


• Finally, we divide both sides by \(0.2\) to solve for \(x\), yielding \(x = \frac{18.19}{0.2}\).

This last calculation gives \(x = 90.95\).
By isolating the variable, we effectively untangle the equation, making it easy to find the solution. This step is a pivotal aspect of solving linear equations as it allows us to solve for the unknown value directly.