Satisfy

Satisfy

To show that substituting one or more variables into an equation or inequality "works out". That is, the equation or inequality simplifies to a true statement.

Example showing x=5 satisfies 4x−12=8 by substituting: 4(5)−12=8 simplifies to 8=8, a true statement.

Example 2 showing x=5 satisfies 4x>8: substituting gives 4(5)=20>8, a true statement.

Example 3 showing that (2,-5) satisfies y²/21 - x²/21 = 1 by substituting x=2, y=-5, simplifying to 21/21=1.

See also

Verify a solution

Key Formula

\text{An equation } f(x) = g(x) \text{ is satisfied by } x = a \text{ if } f(a) = g(a) \text{ is true.}

Where:

$x$ = The variable in the equation or inequality
$a$ = The specific value substituted for x
$f(x), g(x)$ = Expressions on each side of the equation or inequality

Worked Example

Problem: Does x = 3 satisfy the equation 2x + 1 = 7?

Step 1: Write down the original equation.

2x + 1 = 7

Step 2: Substitute x = 3 into the left side of the equation.

2(3) + 1 = 6 + 1 = 7

Step 3: Compare the left side to the right side. Since 7 = 7 is a true statement, the equation is satisfied.

7 = 7 \quad \checkmark

Answer: Yes, x = 3 satisfies the equation 2x + 1 = 7 because substituting 3 produces the true statement 7 = 7.

Another Example

This example differs from the first by involving an inequality instead of an equation, and by substituting two variables (an ordered pair) rather than a single value.

Problem: Does the ordered pair (2, 5) satisfy the inequality 3x + y > 10?

Step 1: Write down the original inequality.

3x + y > 10

Step 2: Substitute x = 2 and y = 5 into the left side.

3(2) + 5 = 6 + 5 = 11

Step 3: Check whether the resulting statement is true.

11 > 10 \quad \checkmark

Step 4: Since 11 > 10 is true, the ordered pair satisfies the inequality.

Answer: Yes, (2, 5) satisfies 3x + y > 10 because substituting gives 11 > 10, which is true.

Frequently Asked Questions

What does it mean to satisfy an equation?

To satisfy an equation means that when you replace the variable(s) with a specific value (or values), both sides of the equation become equal, producing a true statement. For example, x = 4 satisfies x + 6 = 10 because 4 + 6 = 10 is true.

What is the difference between satisfying an equation and solving an equation?

Solving an equation means finding all values that make it true. Satisfying an equation means checking or confirming that a particular value makes it true. You solve first, then you can verify that your answer satisfies the original equation. Every solution satisfies the equation, and every value that satisfies the equation is a solution.

Can more than one value satisfy an equation?

Yes. Some equations have multiple solutions. For example, both x = 3 and x = −3 satisfy the equation x² = 9. Inequalities can have infinitely many values that satisfy them—every x > 2 satisfies 2x > 4.

Satisfy vs. Verify a Solution

	Satisfy	Verify a Solution
Definition	A value satisfies an equation if substituting it produces a true statement	Verifying a solution means checking that a proposed answer satisfies the equation
Direction	Describes the relationship between a value and an equation/inequality	Describes the action a person takes to confirm correctness
When to use	Use when stating that a value makes an equation or inequality true	Use when you have already solved and want to check your answer

Why It Matters

The concept of satisfying an equation appears constantly in algebra, from checking homework answers to testing potential solutions on standardized tests. When you solve a system of equations, you need to confirm that your answer satisfies every equation in the system—not just one. Understanding this idea also prepares you for higher-level topics like satisfying constraints in optimization and determining whether points lie on graphs of functions.

Common Mistakes

Mistake: Substituting into only one side of the equation and forgetting to compare both sides.

Correction: Always evaluate both the left side and the right side after substitution, then check whether the resulting statement (equality or inequality) is true.

Mistake: Concluding that a value satisfies a system of equations after checking only one equation.

Correction: A value (or ordered pair) must satisfy every equation or inequality in the system. Substitute into all of them and confirm each one produces a true statement.

Related Terms

Variable — The unknown you substitute a value for
Equation — A statement that a value can satisfy
Inequality — A comparison statement that values can satisfy
Simplify — Reducing expressions after substitution
Verify a Solution — The process of checking that a value satisfies an equation
Solution — A value that satisfies an equation or inequality
Substitution — The method used to test if a value satisfies
System of Equations — Multiple equations a solution must satisfy simultaneously