Problem 77 - Algebra A Combined Function

Question

Match each equation in the first column with its solution in the second column. Items in the second column may be used more than once. a. all real numbers b. no solution C. 0 $$ 2 x-6 x-10=-4 x+3-10 $$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1: Simplify the Equation

First, we'll simplify both sides of the equation to make it easier to solve. The given equation is:\[ 2x - 6x - 10 = -4x + 3 - 10 \] On the left side, combine like terms:\[ 2x - 6x = -4x \]Thus, the left side becomes:\[ -4x - 10 \]On the right side, combine the constants:\[ 3 - 10 = -7 \]So the equation simplifies to:\[ -4x - 10 = -4x - 7 \]

Step 2: Eliminate the Variable

Next, we want to eliminate \(x\) from both sides if possible to see if there's a solution:Subtract \(-4x\) from both sides:\[ -4x - 10 + 4x = -4x - 7 + 4x \]This simplifies to:\[ -10 = -7 \]

Step 3: Analyze the Result

The result \(-10 = -7\) is a false statement, indicating that the original equation has no solution. The two sides of the equation are not equal, regardless of the value chosen for \(x\). Thus, there is no value of \(x\) that will satisfy the equation.

Key Concepts

No Solution

In the context of linear equations, the term "no solution" refers to scenarios where there is no possible value for the variable that can make the equation true. It means that after simplifying and solving the equation, we end up with a statement that is inherently false, such as \(-10 = -7\).

This occurs when the variable terms cancel each other out, leaving a contradiction. When encountering an equation with no solution, it's important to recognize that this means the lines represented by the equation do not intersect on a graph. They are parallel lines.

Here’s another example to understand further:

• If you have an equation like \( 3x + 5 = 3x + 2 \), simplifying by subtracting \(3x\) from both sides would result in \(5 = 2\), which is clearly impossible. Hence, no solution!

This insight helps in understanding that not all linear equations will yield a solution.

False Statement

A false statement arises in an equation when, after simplifying, both sides contain numeric values that are not equal. This is a crucial checkpoint while solving linear equations.

In our solved example, we reached the statement \(-10 = -7\), which is obviously incorrect. Such an outcome indicates that no numeric value for the variable can satisfy the original equation.

It's akin to claiming that mathematics allows setting unequal values as equal, which disrupts the logical consistency of mathematical truths.

Recognizing a false statement immediately alerts us to the fact that an error occurred during real-world problem setting or that the problem is inherently unsolvable as posed given numerical and variable constraints.

• An important skill in algebra is to identify these false endings, as they swiftly indicate the type of solution (or lack thereof) in an equation.

Combining Like Terms

Combining like terms is a fundamental process during which similar variable terms or constant numbers are consolidated. It helps in simplifying expressions and eventually solving linear equations.

In our original equation, we started with the expression \(2x - 6x - 10\). Combining like terms \(2x\) and \(-6x\) results in \(-4x\). This step reduced the complexity of the original expression. Likewise, grouping constants \(3 - 10\) produced \(-7\) on the right side of the equation.

Here's how you should approach it:

• Identify terms with the same variable and power.
• Add or subtract the coefficients.

After this, your expressions become much easier to work with. Using this technique ensures clarity, minimizes errors, and leads to correct conclusions when solving linear equations.