Problem 23 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$7=3(x+2)-3(x-5)$$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1: Distribute

Apply the distributive property on the equation. This will result in \(7=3x+6-3x+15\)

Step 2: Simplify

Simplify the equation by combining like terms on the right side of the equation. This results in \(7=21\)

Step 3: Check for solution

We notice that the simplified equation \(7=21\) has no x term and that the equation is not valid, which means there is no solution for the initial problem.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify equations by breaking down expressions within parentheses. We apply the distributive property by multiplying each term inside the parentheses by the factor outside. This helps to make equations easier to solve by eliminating parentheses.

For example, in the equation provided, we were given:

• \( 3(x+2) \) which, using the distributive property, becomes \( 3 \times x + 3 \times 2 = 3x + 6 \).
• Similarly, \( -3(x-5) \) becomes \( -3 \times x + 15 = -3x + 15 \).

Breaking down these expressions makes it easier to see and combine like terms in later steps, as we deal with simpler expressions instead of complex ones at the start.

Simplifying Equations

Simplifying equations is an important step in solving algebraic problems as it helps to condense expressions and make them easier to work with. In simplification, we aim to combine like terms and reduce expressions to a simpler form.

After applying the distributive property in our given problem, we were left with:

• \( 7 = 3x + 6 - 3x + 15 \).

The next step is to combine like terms. Here, the terms \(3x\) and \(-3x\) cancel each other out, leaving us with:

• \( 7 = 21 \).

Reducing the equation helps us identify contradictions or lead to further calculations. Simplifying accurately saves time and aids in recognizing if a solution exists.

Solving Equations with No Solution

Sometimes, while solving equations, we arrive at a point where the equation is no longer valid—it produces a statement that is false, such as \( 7 = 21 \). This indicates that the equation has no solution, meaning no value of x will satisfy the original equation.

When simplified equations lead to a false statement, it highlights inconsistencies. It's crucial to recognize these situations as they show that the original equation was set in a way that no numbers satisfy the given condition.

Therefore, in problems where you end up with an impossible statement, it's correct to conclude that there is no solution. This concept is a reminder that not all mathematical equations have solutions, and recognizing these cases is important when solving algebraic problems.