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Chapter 2: Problem 63

Solve each equation, and check the solution. $$ 6 r-8 r=10 $$

Short Answer

Expert verified
The solution is \(r = -5\).

Step by step solution

01

Combine like terms

First, combine the like terms on the left side of the equation. The equation is given by: \(6r - 8r = 10\). Combine the terms with the variable \(r\) to get: \(-2r = 10\).
02

Isolate the variable

Next, we need to solve for \(r\). Divide both sides of the equation by -2 to isolate \(r\): \(r = \frac{10}{-2}\).
03

Simplify the solution

Simplify the right side of the equation to find the value of \(r\): \(r = -5\).
04

Check the solution

Substitute \(r = -5\) back into the original equation to verify the solution: \(6(-5) - 8(-5) = 10\). Simplify both terms: \(-30 + 40 = 10\). Since the equation holds true, the solution \(r = -5\) is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
To solve linear equations, we often deal with terms that have the same variable. These terms are known as 'like terms'. The first step in solving the given equation is to combine these like terms. In the equation provided, we have two terms involving the variable 'r': \(6r - 8r = 10\).

To combine them, simply subtract the coefficients (the numbers in front of the variable), which results in \(-2r\). Now the equation looks simpler: \(-2r = 10\).

Combining like terms helps in reducing the complexity of an equation, making it easier to find the solution.
Isolating Variables
After combining like terms, the next step is to isolate the variable. In our case, we have the equation \(-2r = 10\). Our goal is to solve for 'r'.

To isolate 'r', we need to 'undo' the multiplication of -2. We do this by dividing both sides of the equation by -2. Here鈥檚 the calculation: \(r = \frac{10}{-2}\).

This simplifies to:\(r = -5\).

Isolating the variable means getting it alone on one side of the equation, which makes it easy to solve.
Checking Solutions
Once we have a solution, it's important to verify it. We do this by substituting the solution back into the original equation. This check ensures that we have the correct answer.

For our equation, we substitute \(r = -5\) back into \(6r - 8r = 10\). This gives us: \(6(-5) - 8(-5) = 10\). Performing the multiplication, we get: \(-30 + 40 = 10\).

The left side simplifies to 10, which matches the right side of the equation. Therefore, the solution \(r = -5\) is verified to be correct.

Always remember to check your solutions to avoid any mistakes in your calculations.

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Most popular questions from this chapter

The sum of the measures of the angles of any triangle is \(180^{\circ} .\) In triangle \(A B C,\) the measure of angle \(A\) is \(141^{\circ}\) greater than the measure of angle \(B\). The measure of angle \(B\) is the same as the measure of angle \(C\). Find the measure of each angle.

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