Problem 44 - Technical Mathematics with Calculus

Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. Equations Having Symbols of Grouping. $$5(2 x-8)=2(10-5 x)$$

Step-by-Step Solution

Answer

The solution for the equation is \(x = \frac{20}{20} = 1\). After checking, we see that the value of x satisfies the original equation.

Step 1: Distribute

Use the distributive property to eliminate the parentheses by multiplying the coefficients outside the parenthesis by the terms inside. On the left side of the equation, multiply 5 by both 2x and -8. On the right side, multiply 2 by both 10 and -5x.

Step 2: Simplify both sides

After distributing the coefficients, combine like terms on both sides of the equation. The left side becomes 10x - 40, and the right side becomes 20 - 10x.

Step 3: Move variable terms to one side

Add 10x to both sides to get all the x terms on one side and the constants on the other side of the equation.

Step 4: Combine like terms

Combine the x terms and the constant terms separately. You have 20x on the left side and 20 on the right side after combining like terms.

Step 5: Isolate the variable

Divide both sides of the equation by the coefficient of x, which is 20, to solve for x.

Step 6: Check the solution

Substitute the obtained value of x back into the original equation to verify if the equation holds true.

Key Concepts

Distributive Property

The distributive property is a cornerstone algebraic concept used for solving equations, especially when they involve parentheses. In an equation, if you have a term outside the parentheses multiplied by the terms inside it, you must apply the distributive property to simplify the equation. For instance, in the problem \(5(2x-8)=2(10-5x)\), the coefficients 5 and 2 are distributed over the terms within the parentheses.

Here’s how you do it:

• For the left side: Multiply 5 by both 2x and -8 to get \(5\times 2x - 5\times 8\).
• For the right side: Multiply 2 by 10 and -5x to get \(2\times 10 - 2\times 5x\).

By following this method, you eliminate the parentheses and prepare the equation for the next steps:

Combining Like Terms

Combining like terms further simplifies the equation. Terms are 'like' if they have the same variables raised to the same power. To combine them, simply add or subtract their coefficients. After distributing, you often find like terms both within single sides of an equation and across the equation.

In the example, \(10x - 40 = 20 - 10x\), you have \(10x\) and \(10x\) as like terms, as well as the numbers -40 and 20. Here's what you should do:

• Add \(10x\) from the right side to the left to centralize the variable terms: \(10x + 10x\).
• Combine the constants on one side: -40 and 20.

By doing so, the equation is further simplified, enabling easier isolation of the variable.

Isolating the Variable

Isolating the variable, often the final step in solving linear equations, involves manipulating the equation to have the variable term on one side and the constant term on the other. You aim to get the variable by itself—hence the term 'isolated.'

In our problem, after combining like terms, we get \(20x - 40 = 20\). To isolate \(x\), you must get rid of the number that is attached to it, which is the coefficient 20:

• First, add 40 to both sides to balance the constants: \(20x = 60\).
• Then divide everything by the coefficient of \(x\), which is 20, so you have \(x = \frac{60}{20}\).

Once completed, \(x\) stands alone with its value clearly determined.

Checking Equations

After solving an equation, it’s crucial to check your solution to ensure accuracy. This involves substituting the solution back into the original problem and verifying if both sides of the equation balance.

For example, if you found \(x = 3\) in the solved equation, plug \(x\) back into the original equation: \(5(2\times\frac{3}{1}-8) = 2(10 - 5\times\frac{3}{1})\). Simplify both sides to see if they equal the same value. Here’s how:

• Calculate the left side: \(5(6 - 8) = 5(-2) = -10\).
• Calculate the right side: \(2(10 - 15) = 2(-5) = -10\)

If both sides are equal after the substitution, your solution is correct. In this case, they are equal, confirming \(x = 3\) as the right answer.