Problem 75 - Algebra and Trigonometry

Question

\(71-78\) Find the solution of the equation correct to two decimals. $$ 3.16(x+4.63)=4.19(x-7.24) $$

Step-by-Step Solution

Answer

The solution is approximately \(x = 43.67\).

Step 1: Expand Both Sides

First, we expand both sides of the equation. Distribute the constants over the terms inside the parentheses:Left Side: \[3.16(x + 4.63) = 3.16x + 3.16 imes 4.63 = 3.16x + 14.6408\]Right Side:\[4.19(x - 7.24) = 4.19x - 4.19 imes 7.24 = 4.19x - 30.3356\]

Step 2: Rearrange the Equation

Move all the terms involving \(x\) to one side of the equation and constant terms to the other side:Starting from:\[3.16x + 14.6408 = 4.19x - 30.3356\]Rearrange to gather the \(x\) terms on one side:\[3.16x - 4.19x = -30.3356 - 14.6408\]

Step 3: Simplify the Equation

Simplify both sides by combining like terms:Combine \(x\) terms:\[-1.03x = -44.9764\]

Step 4: Solve for \(x\)

Divide both sides of the equation by -1.03 to isolate \(x\):\[x = \frac{-44.9764}{-1.03} = 43.667\]

Step 5: Round the Solution

Round the solution to two decimal places as required by the problem:\[x \approx 43.67\]

Key Concepts

Distribution in Algebra

Distribution in algebra is all about multiplying a single term by each term inside a parenthesis. It’s as if you’re sharing the outside number with everything inside.
Think of it like distributing snacks from a box to your friends: each friend gets the same portion. Mathematically, it’s shown as:

• Given: \( a(b + c) \)
• Distribute: \( ab + ac \)

In our exercise, we use distribution to expand both the left and right sides of the equation. For instance, on the left side:

• \( 3.16(x + 4.63) = 3.16 \times x + 3.16 \times 4.63 = 3.16x + 14.6408 \)

This helps to separate each element neatly, making the equation easier to handle. It's a foundational skill in algebra that you'll use often.

Combining Like Terms

Once you’ve used the distribution, the next step is to make the equation simpler by combining like terms. Like terms are kind of like siblings—they look similar because they have the same variable or are constants.
In mathematics, this would mean you’re combining terms like:

• \( ax + bx = (a+b)x \)
• \( c + d = (c+d) \)

In our example, after distributing, we rearrange the equation to group similar terms. We see terms with \( x \) on both sides:

• \( 3.16x - 4.19x \)

These get grouped together into a single term by subtraction, which gives \( -1.03x \). This step makes the equation much more straightforward and manageable.

Rearranging Equations

Rearranging equations is an important process in algebra where we shift things around to isolate the variable we're solving for.
It’s like tidying up: you want the variable on one side and everything else on the other. Steps include:

• Moving variable terms to one side of the equation
• Moving constant terms to the opposite side

This often involves adding, subtracting, or even multiplying/dividing every term by the same number to maintain balance. For instance:

• Starting from: \( 3.16x + 14.6408 = 4.19x - 30.3356 \)
• We rearrange to: \( 3.16x - 4.19x = -30.3356 - 14.6408 \)

This neat separation makes it easier to solve the equation, like finally finding an answer inside an unjumbled context!

Rounding Decimals

After you've solved for the variable, the final touch is sometimes rounding off to the desired number of decimal places. Rounding is crucial when expressing your answer neatly or fitting your result to the problem's requirement.
Here's how you do it:

• If the number is greater than 5, round up
• If less than 4, round down
• If exactly 5, round to the nearest even number for fairness in statistics

From our exercise, the solution was \( x = 43.667 \), but it needed to be rounded to two decimal places. Applying the rules:

• Since the thousandths place is 7, we round up: \( x \approx 43.67 \)

This finishes the solving process, ensuring a precise but clear presentation of your result.