Problem 68 - Introductory Algebra for College Students

Question

Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$5 x-3(x+1)=2(x+3)-5$$

Step-by-Step Solution

Answer

The equation has no solution as it equals to a false mathematical statement, \(-3 = 1\).

Step 1: Distribute

First, let's distribute the numbers outside the parentheses to simplify the equation. We get \(5x - 3x - 3 = 2x + 6 - 5\).

Step 2: Simplify Both Sides of the Equation

Now simplify both sides of the equation by combining like terms to make the equation easier to solve. This gives us \(2x - 3 = 2x + 1\).

Step 3: Solve for x

Our next step is to solve for \(x\). To do so, we aim to isolate \(x\) on one side of the equation. But we observe that if we subtract \(2x\) from both sides, the \(x\) terms will cancel out on both sides giving us \(-3 = 1\).

Step 4: Analysis of the Result

After attaining \(-3 = 1\), we notice that this is an incorrect mathematical statement which informs us that the original equation has no solution.

Key Concepts

Distributive Property

When solving linear equations, one of the primary tools is the distributive property. This property helps us to break down expressions in parentheses by distributing a factor across terms within the parentheses. For example, in the equation \(5x - 3(x + 1) = 2(x + 3) - 5\), the distributive property is applied to both \(-3(x + 1)\) and \(2(x + 3)\).

By applying the distributive property, we simplify the equation as follows:

• \(-3(x + 1)\) becomes \(-3x - 3\).
• \(2(x + 3)\) becomes \(2x + 6\).

This simplification is crucial because it reduces the number of terms and makes the equation easier to work with. Understanding this property is essential in solving more complex equations.

Combining Like Terms

After using the distributive property, the next step is simplifying the algebraic expression by combining like terms. Like terms are terms that have the same variables raised to the same power. In our example, after distribution, we get the equation \(5x - 3x - 3 = 2x + 6 - 5\).

To combine like terms:

• Look for terms with the same variable, like \(5x\) and \(-3x\), and combine them to get \(2x\).
• Combine the constant terms, such as \(-3\) and \(-5\), to simplify the equation further.

The simplified equation \(2x - 3 = 2x + 1\) highlights how combining like terms helps clear up the equation, making it easier to understand and solve.

No Solution Equations

As we progress in solving the equation, sometimes we encounter a situation where, after simplifying, the variables vanish and we are left with a false statement. This results in what is known as a "no solution" equation. In our example, after combining and attempting to isolate the variable \(x\), we ended up with the statement \(-3 = 1\).

This situation indicates a contradiction:

• No value of \(x\) exists that will satisfy the equation \(-3 = 1\).
• Equations that result in false statements are said to have no solution.

Identifying no solution equations is important because it saves time and ensures accuracy in mathematical problem-solving processes.

Isolation of Variables

The process of solving linear equations usually involves isolating the variable to one side of the equation to find its value. However, in some scenarios, such as in the current example, isolating the variable terms can lead to discovering that the equation has no solution.

Isolating the variable involves:

• Moving all terms involving the variable to one side of the equation.
• Balancing constant terms on the opposite side.

In the case \(2x - 3 = 2x + 1\), attempting to isolate \(x\) by subtracting \(2x\) from both sides results in the cancellation of \(x\) terms, which leaves us with a false statement. This approach shows how variable isolation can reveal more about the structure and solution set of an equation.