Problem 55 - Prealgebra

Question

Solve the equation. \(-1.8(-1.6 x+1.7)=-1.8(-3.6 x-4.1)\)

Step-by-Step Solution

Answer

The solution is \(x = -2.9\).

Step 1: Distribute on the Left Side

First, distribute the \(-1.8\) on the left-hand side of the equation. This means multiplying \(-1.8\) by each term inside the parenthesis: \(-1.8 \times -1.6x + (-1.8 \times 1.7)\).

Step 2: Distribute on the Right Side

Now, apply distribution on the right-hand side of the equation by multiplying \(-1.8\) with \(-3.6x\) and \(-4.1\). This results in: \(-1.8 \times -3.6x + (-1.8 \times -4.1)\).

Step 3: Calculate Products

Calculate each multiplication obtained in the previous steps: - Left: \(-1.8 \times -1.6x = 2.88x\) and \(-1.8 \times 1.7 = -3.06\)- Right: \(-1.8 \times -3.6x = 6.48x\) and \(-1.8 \times -4.1 = 7.38\).

Step 4: Combine Terms on Each Side

Substitute the products back into the equation:- Left: \(2.88x - 3.06\)- Right: \(6.48x + 7.38\)The complete equation becomes \(2.88x - 3.06 = 6.48x + 7.38\).

Step 5: Rearrange Equation

To solve for \(x\), rearrange the equation. First, move all terms involving \(x\) to one side by subtracting \(2.88x\) from both sides: \(2.88x - 3.06 - 2.88x = 6.48x + 7.38 - 2.88x\).

Step 6: Simplify Equation

Simplify the equation to get all constant terms on the other side by subtracting 7.38 from both sides:\(-3.06 - 7.38 = 6.48x - 2.88x\).

Step 7: Solve for x

After simplification, continue solving the equation:- \-10.44 = 3.60x\- Divide both sides by 3.60 to find \(x\):\(x = \frac{-10.44}{3.60}\).

Step 8: Final Step: Calculate Value of x

Calculate \(x\) by performing the division:\(x = -2.9\).

Key Concepts

Distributive Property

When solving equations like the one provided, the distributive property is a vital tool. This property allows us to simplify expressions by multiplying a single term by each term inside a set of parentheses. In this problem, we started with \(-1.8(-1.6x + 1.7)\) on the left side. Here, using the distributive property, you multiply \(-1.8\) by each term inside the parentheses, which are \(-1.6x\) and \(1.7\).

• This gives us two products: \(-1.8 imes -1.6x = 2.88x\)
• and \(-1.8 imes 1.7 = -3.06\).

The same property applies to the right side of the equation. By applying it, we manipulate the expression so it becomes simpler and each term can be clearly seen and calculated.
Understanding this property helps break down complex expressions in algebra into manageable parts.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to solve for unknowns. In this context, once we've distributed and simplified the terms using the distributive property, our next step is to combine like terms if possible. It’s crucial to keep the equation balanced, which means any operation done on one side must be done on the other.For our equation, after distribution, we had \(2.88x - 3.06 = 6.48x + 7.38\). Algebraic manipulation involves moving \(2.88x\) to the other side:

• Subtract \(2.88x\) from both sides to keep the balance.
• This leaves us with only the constant and variable terms on each side.

Sometimes, you might need to add, subtract, or both, different values to streamline the expression further, which eventually aids in isolating the variable.

Linear Equations

A linear equation is any equation that forms a straight line when graphed. Such equations usually have constants and variables that are raised to the power of one, which makes them easier to manipulate and solve. The equation in our example is in its linear form, meaning the highest power of the variable, \(x\), is one. After simplifying, our equation looks like \(-3.06 = 6.48x - 2.88x + 7.38\). Linear equations can be solved with straightforward algebraic steps, unlike quadratic or cubic equations that involve terms like \(x^2\) or \(x^3\).

• By consolidating all \(x\) terms and constants, you are essentially moving towards isolating \(x\) on one side of the equation.
• Then compute the necessary calculations to reach a solution.

Fully understanding linear equations also forms a foundation for dealing with more complex mathematical problems.

Mathematical Calculations

Finally, in solving equations, performing mathematical calculations accurately is crucial. This process involves computing products, sums, and differences carefully to avoid mistakes that propagate through the solving process.In our example, once we've rearranged the equation to \(-10.44 = 3.60x\), it becomes a matter of calculation to solve for \(x\). At this point:

• Divide both sides by \(3.60\) to isolate \(x\).
• This calculation results in \(x = \frac{-10.44}{3.60} = -2.9\).

Here, attention to numerical accuracy is paramount. Every mistake can lead to an incorrect solution, so it is important to double-check each calculation.