Problem 18 - Prealgebra

Question

Solve each equation. $$6 n+0.88=2 n-0.77$$

Step-by-Step Solution

Answer

The solution is \( n = -0.4125 \).

Step 1: Remove Constant on Right

We start by moving the constant term from the right side to the left side of the equation. This involves adding 0.77 to both sides:\[6n + 0.88 + 0.77 = 2n - 0.77 + 0.77\]Simplifying, we get:\[6n + 1.65 = 2n\]

Step 2: Eliminate n from Right Side

Next, we eliminate the variable term on the right side by subtracting \(2n\) from both sides:\[6n + 1.65 - 2n = 2n - 2n\]This simplifies to:\[4n + 1.65 = 0\]

Step 3: Isolate the Variable

Now, we need to isolate \(n\). Subtract 1.65 from both sides to remove the constant term:\[4n = -1.65\]

Step 4: Solve for n

Finally, we solve for \(n\) by dividing both sides by 4:\[n = \frac{-1.65}{4}\]Calculating this division, we find:\[n = -0.4125\]

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental concept in solving linear equations. It involves the reshuffling and rearranging of terms within an equation to simplify and solve it. In essence, it's about strategically moving things around using mathematical operations while ensuring that the equality of the equation remains valid.
When manipulating, keep these tips in mind:

• Use addition or subtraction to eliminate constant terms on one side.
• Apply multiplication or division to tweak coefficients (the numbers in front of variables).

For example, in the equation provided \[6n + 0.88 = 2n - 0.77\] our first algebraic manipulation is to balance the constants by adding or subtracting numbers. This process aids in simplifying the equation to a form that’s easier to work with. Simple arithmetic steps like this make complex-looking equations much friendlier to solve.

Equation Balancing

Balancing an equation is similar to keeping a scale even. Whatever operation you do on one side must be done to the other to maintain equality. This concept is pivotal for ensuring the integrity of the equation throughout the solving process.
To start balancing the given equation, \[6n + 0.88 = 2n - 0.77\], we aim to simplify by aligning similar terms across the equation. Because both sides should have equivalent values, any operation such as adding, subtracting, multiplying, or dividing needs to be mirrored:

• Add 0.77 to eliminate -0.77, balancing the constant terms.
• Subtract \(2n\) from both sides to neutralize the variable on one side of the equation.

These operations carve the way for isolating the variable, leading us to a simpler equation. Remember, balance is key in turning a complicated equation into a straightforward one.

Variable Isolation

Variable isolation is a technique used to "free" the variable from surrounding terms in order to solve for it. It involves removing constants and coefficients so that the variable stands alone on one side of the equation.
In our example, after simplifying and balancing the equation, we have: \[4n + 1.65 = 0\].The next step involves isolating \(n\) by getting rid of any numbers that are lumped with it. Start by subtracting 1.65 from both sides, leaving:\[4n = -1.65\].Once you've cleared the constant, proceed by dividing through by 4, the coefficient of the variable:\[n = \frac{-1.65}{4}\].
By isolating the variable, we not only simplify the problem but also arrive directly at its solution. This method ensures precise solutions and reinforces mastery in solving equations.