Algebraic equations with no answer: A clear explanation for high school students

📚 Understanding Algebraic Equations with No Solution

In algebra, we're often trying to find the value of a variable (usually $x$) that makes an equation true. Sometimes, no matter what value we plug in for $x$, the equation will *never* be true. These are equations with no solution. Let's break it down!

💡 What Does 'No Solution' Really Mean?

Imagine an equation like this: $x + 2 = x + 3$.

We want to find a value for $x$ that makes both sides of the equation equal. If we try to solve it, we might subtract $x$ from both sides:

$x + 2 - x = x + 3 - x$

This simplifies to:

$2 = 3$

Wait a minute... $2$ does NOT equal $3$! This is a contradiction. No matter what number we choose for $x$, the equation will always be false. That's because adding the same value ($x$) to both sides won't change the difference between the two sides (2 and 3). The equation has no solution!

📝 How to Identify Equations with No Solution

• ⚖️ Simplification leads to a contradiction: As shown above, if you simplify the equation and end up with a statement that is always false (like $2 = 3$), then the equation has no solution.
• 📉 Variables cancel out: When solving, the variable terms ($x$ terms) completely disappear, leaving you with a false statement.
• 🚧 Parallel Lines (Visually): Think about linear equations as lines. If the lines represented by the two sides of the equation are parallel, they will never intersect, meaning there's no single $x$ value that satisfies the equation. This is equivalent to 'no solution'.

➗ Example Problems

Let's try a few examples:

1. Solve for $x$: $2(x + 1) = 2x + 5$

   Expand: $2x + 2 = 2x + 5$

   Subtract $2x$ from both sides: $2 = 5$ (No solution!)

2. Solve for $x$: $3x - 4 = 3x + 1$

   Subtract $3x$ from both sides: $-4 = 1$ (No solution!)

3. Solve for $x$: $5(x - 2) = 5x - 10$

   Expand: $5x - 10 = 5x - 10$

   Subtract $5x$ from both sides: $-10 = -10$ (This is always true – infinitely many solutions, not 'no solution'!)

🧪 Practice Quiz

Determine if each of the following equations has no solution, or infinitely many solutions.

1. $4x + 3 = 4x - 1$
2. $2(x - 3) = 2x - 6$
3. $7x + 5 = 7x + 5$
4. $-3x + 2 = -3x + 7$