Equations with No Solution or Infinite Solutions

Three Possible Outcomes

When you solve a linear equation, you reach one of three outcomes:

Outcome	What Happens	Example
One solution	You get $x = \text{a number}$	$2x + 3 = 7 \Rightarrow x = 2$
No solution	You get a false statement	$2x + 3 = 2x + 7 \Rightarrow 3 = 7$
Infinite solutions	You get a true statement	$2x + 3 = 2x + 3 \Rightarrow 3 = 3$

No Solution (Contradiction)

An equation has no solution when simplifying leads to a statement that is always false, such as $0 = 5$ or $3 = -1$.

This happens when both sides have the same variable terms but different constants.

Example: $3x + 4 = 3x + 9$

Subtract $3x$: $4 = 9$ ✗ — this is false, so there is no solution.

Graphically, this corresponds to two parallel lines that never intersect.

Infinite Solutions (Identity)

An equation has infinitely many solutions when simplifying leads to a statement that is always true, such as $0 = 0$ or $5 = 5$.

This happens when both sides are identical after simplification.

Example: $2(x + 3) = 2x + 6$

Expand: $2x + 6 = 2x + 6$

Subtract $2x$: $6 = 6$ ✓ — this is always true, so there are infinitely many solutions.

Graphically, this is the same line drawn twice — every point is an intersection.

Finding a Constant for No Solution

SAT questions often ask: "For what value of $k$ does the equation have no solution?"

Example: For what value of $k$ does $3x + 5 = kx + 8$ have no solution?

For no solution, the variable terms must be equal (coefficients of $x$ match) but the constants must differ.

Coefficients match when $k = 3$. Then: $3x + 5 = 3x + 8$, which gives $5 = 8$ — no solution.

Answer: $k = 3$

Finding a Constant for Infinite Solutions

Example: For what value of $a$ does $a(x + 2) = 3x + 6$ have infinitely many solutions?

Expand: $ax + 2a = 3x + 6$

For infinite solutions, both sides must be identical:

• Coefficient of $x$: $a = 3$
• Constant: $2a = 6 \Rightarrow a = 3$ ✓

Answer: $a = 3$

Checking Both Conditions

When a question involves a parameter and asks for "no solution" or "infinite solutions", you typically need to:

1. Match the coefficients of the variable terms.
2. Check whether the constant terms also match (infinite solutions) or don't match (no solution).

Tip 1: Always Simplify Completely

Distribute and combine like terms on both sides. Only then can you tell whether the equation is a contradiction, identity, or regular equation.

Tip 2: Compare Coefficients

Once simplified to the form $ax + b = cx + d$:

• If $a \neq c$: one solution ($x = \frac{d - b}{a - c}$)
• If $a = c$ and $b \neq d$: no solution
• If $a = c$ and $b = d$: infinitely many solutions

Tip 3: On the SAT, Look for Matching Variable Terms

If you see the same $x$-coefficient on both sides of the equation, immediately check the constant terms. That tells you whether it's no solution or infinite solutions.

Tip 4: Plug In the Given Constant

For questions asking "what value of $k$ gives no solution", substitute your candidate value back in and verify the equation indeed has no solution.

Tip 5: Watch for Tricky Formatting

Sometimes the SAT writes these as multiple-choice questions where one answer gives no solution and another gives infinite solutions. Read carefully.