Problem 60 - Elementary Algebra

Question

Solve. $$ -2(5 x-3)-1=5(-2 x+1) $$

Step-by-Step Solution

Answer

The solution is all real numbers; the equation is an identity.

Step 1: Distribute the Coefficients

First, apply the distributive property to expand the expression on both sides of the equation.The left side: \[-2(5x - 3) = -2 \cdot 5x + (-2) \cdot (-3) = -10x + 6\]Subtract 1 from this result:\[-10x + 6 - 1 = -10x + 5\]The right side:\[5(-2x + 1) = 5 \cdot (-2x) + 5 \cdot 1 = -10x + 5\]

Step 2: Simplify Both Sides

Equate both expanded expressions from the previous step:\[-10x + 5 = -10x + 5\]

Step 3: Analyze the Equation

Notice that both sides of the equation are identical \[-10x + 5 = -10x + 5\]This indicates that the equation is always true for any value of \(x\).

Step 4: Determine the Solution Set

Since the equation is true for any \(x\), we conclude that the solution is all real numbers.

Key Concepts

Distributive Property

The distributive property is a powerful tool in algebra that allows you to multiply a single term by a group of terms inside parentheses. It's much like spreading out a multiplier across terms that are added or subtracted inside the brackets. For example, if we look at the expression \(-2(5x - 3)\), the distributive property helps us expand it as follows:

• Multiply \(-2\) by \(5x\) to get \(-10x\).
• Multiply \(-2\) by \(-3\) to get \(+6\).

Then, when we put it all together, the expression becomes \(-10x + 6\). You do the same thing with the right side of the equation \(5(-2x + 1)\) which expands to \(-10x + 5\).
This property helps not just to expand expressions, but to transform complex equations into simpler ones, ready for further solving steps.

Simplifying Equations

Simplifying equations is like tidying up your room. We take complex expressions and make them easier to work with by combining like terms and removing unnecessary operations. In our exercise, after using the distributive property, we simplify the equation \(-10x + 5 = -10x + 5\). Notice that both sides of the equation are already equal and straightforward.

• No extra terms are hanging around.
• Neither side appears more complex than the other.

At this point, the equation becomes simply that each side mirrors the other. Simplifying it further, you see that everything is in place, meaning every part of the equation is aligned properly.
You cannot eliminate terms without altering the balance, because it's already perfectly balanced. This realization is a key step in understanding equalities.

Solution Set

The solution set of an equation tells you what values of the variable will make the equation true. In our equation \(-10x + 5 = -10x + 5\), we found that both sides are exactly the same.
This indicates that no matter what value you assign to \(x\), the equation holds true. This is a special kind of solution.

• The equation is called an identity.
• The solution set is all real numbers.

This happens because the equation doesn’t restrict \(x\) at all - it's valid across the whole number line. Understanding that such scenarios exist helps in distinguishing when specific solutions or broader ones apply in real-world situations.