Problem 21 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$-6(3+x)+2(3 x+5)=0$$

Step-by-Step Solution

Answer

The given equation has no solution because the final equation -8 = 0 is a false statement.

Step 1: Distribute

First, distribute the -6 to both 3 and x within the brackets and do the same with 2 to 3x and 5 in the other bracket. By doing this we obtain: \[-18 - 6x + 6x +10 = 0\]

Step 2: Simplify

Now, combine like terms. Add the constants (-18+10) and the x terms (-6x and +6x) on the left side of the equality. So, the equation becomes: \[-8 = 0\]

Step 3: Check For A Solution

Looking at `-8 = 0`, we can see that it's saying -8 is equal to 0, which is clearly a false statement. No matter how we manipulate it, it will always be false because -8 can't be equal to 0. Therefore, this equation has no solution.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that helps us simplify and solve equations. It states that when you multiply a number by a sum, you can distribute the multiplication across each addend. This is formally expressed as:

• \(a(b + c) = ab + ac\)

In the given equation \(-6(3+x)+2(3x+5)=0\), the distributive property allows us to remove the parentheses. We start by distributing \(-6\) across \(3\) and \(x\). So, \(-6 \times 3 = -18\) and \(-6 \times x = -6x\).
Similarly, for the \(2(3x+5)\) part, distribute \(2\) to both \(3x\) and \(5\). This calculation gives us \(2 \times 3x = 6x\) and \(2 \times 5 = 10\).
The result after applying the distributive property is:

• \(-18 - 6x + 6x + 10\)

By understanding this, you can confidently simplify many algebraic expressions and equations in your math journey.

Combining Like Terms

After distributing, the next key step in solving an equation is combining like terms. This practice involves simplifying expressions by adding or subtracting the coefficients of terms that share the same variables or constants.In our equation, after using the distributive property, we arrive at:

• \(-18 - 6x + 6x + 10\)

Notice how \(-6x\) and \(+6x\) have the same variable \(x\). We can simplify them by adding the coefficients (the numbers in front of \(x\)).
Here, \(-6 + 6 = 0\), so the \(x\) terms cancel each other out, leaving us with:

• \(-18 + 10\)

This part simplifies further by combining constants, \(-18 + 10 = -8\).
With this step complete, you now have a simpler equation to analyze, which helps determine the solvability of an equation.

Equation Solving Steps

Solving an equation involves a series of methodical steps that ensure logic and structure in approach. Initially, we use techniques such as the distributive property and combining like terms to simplify the equation. After simplification, we proceed to analyze the remaining equation for potential solutions.In the equation provided, the simplification gives us

• \(-8 = 0\)

This is a peculiar step where interpretation is crucial. Notice how \(-8\) is incorrectly equated to \(0\). In mathematics, this direct contradiction signifies that no solution exists, as no value will satisfy this false statement.
Recognizing such scenarios prevents futile efforts in finding solutions where none exist.By following systematic steps:

• Apply distributive property
• Combine like terms
• Analyze the simplified statement

you will enhance accuracy and understanding in problem-solving tasks.