Question

Solve the equation if possible. $$ 6-(-5 r)=5 r-3 $$

Short answer

The equation \( 6 - (-5r) = 5r - 3 \) has no solution because it leads to the contradiction \( 6 = -3 \).

Step-by-step solution

Simplify the Equation

The equation given is \( 6 - (-5r) = 5r - 3 \). Start by simplifying the equation. The term \( -(-5r) \) becomes \( +5r \). Thus, the equation becomes \( 6 + 5r = 5r - 3 \).

Combine Like Terms

Next, combine like terms. Subtract \( 5r \) from both sides of the equation to get rid of \( 5r \) on the right-hand side. This gives: \( 6 = -3 \).

Conclusion

The equation \( 6 = -3 \) is a contradiction. This means there are no solutions to the original equation, because no matter what value we plug in for \( r \), it will not make the original equation true.

Key concepts

Simplify Equations

When you encounter a linear equation, the first step is often to simplify it. Simplifying might sound a little vague, but it has a very specific meaning in algebra. Essentially, you're making the expression easier to work with. This process typically involves several components:

• Removing parentheses by applying the distributive property or by simply acknowledging that a double negative turns into a positive, as with the term \( -(-5r) \).
• Combining any constants or simplifying basic arithmetic. For instance, in our problem, there are no constants attached with \( r \), so this becomes just eliminating the confusion of the double negative to get \( 6 + 5r \).
• Making sure you deal with any coefficients (numbers in front of variables) in a clear manner, always remembering that the sign before a term is part of that term.

These steps allow us to take a possibly convoluted equation and turn it into something more straightforward. This is crucial, as it makes the rest of our algebraic work much, much easier.

Combining Like Terms

The term like terms refers to terms in an equation that have the exact same variable component. These can (and should!) be combined to simplify the equation further. In the context of our problem, we're looking at the term \( 5r \) on both sides of the equation. Because they're like terms, we can subtract \( 5r \) from both sides to make them cancel out.

Once you understand the concept, combining like terms is quite straightforward:

• Only terms with the exact same variable parts can be combined.
• Coefficients are added or subtracted while the variable part remains untouched.
• If there are no coefficients, it's just a simple addition or subtraction of numbers.

Our given exercise has a clear application of this: by subtracting \( 5r\) from each side, we're using the principle that things equal to the same thing are equal to each other, thus maintaining the balance of the equation.

No Solution Contradiction

Algebra is mostly about finding values for variables that make an equation true. However, sometimes we find that no value will satisfy the equation. This is known as a contradiction, where the equation simplifies to a statement that's obviously false, such as \( 6 = -3 \).

In these cases, we say the equation has no solution because there's no real number that we can substitute for the variable to make the equation true. Here’s what to bear in mind:

• A contradiction occurs when two equivalent expressions are proven to be unequal after simplification.
• When you reach a false statement like our example, it indicates that the original equation has no solution.
• It's important to recognize this so you don't waste time trying to find a solution where there isn't one.

Learning to spot these contradictions can save you a lot of time and effort, and understanding that some equations have no solution is an essential part of algebra.