Example Of An Equation With One Solution

Understanding equations with one solution is fundamental in algebra and essential for problem-solving in mathematics. Whether analyzing linear relationships, quadratic functions, or systems of equations, recognizing scenarios where a single answer exists helps build a strong foundation for advanced topics. This article explores examples of equations with exactly one solution, explains their characteristics, and provides step-by-step solutions to reinforce comprehension.

Linear Equations: The Simplest Case

Linear equations in one variable are the most straightforward examples of equations with one solution. These equations follow the form ax + b = 0, where a and b are constants, and a ≠ 0. When solved, they yield a single value for x.

Example 1: Solve the equation 2x + 3 = 7.

1. Subtract 3 from both sides:
   2x = 7 – 3
   2x = 4
2. Divide both sides by 2:
   x = 4/2
   x = 2

Here, x = 2 is the only solution. Graphically, this represents the point where the line y = 2x + 3 intersects the x-axis.

Example 2: Solve 5x – 10 = 0.

1. Add 10 to both sides:
   5x = 10
2. Divide by 5:
   x = 2

Again, the solution is unique. g.Linear equations always have one solution unless they are contradictory (e.Also, g. , 0x = 5) or identities (e., 0x = 0), which are exceptions Still holds up..

Quadratic Equations: When the Discriminant Equals Zero

Quadratic equations, of the form ax² + bx + c = 0, can have one, two, or no real solutions depending on the discriminant b² – 4ac. If the discriminant is zero, the equation has exactly one real solution (a repeated root) The details matter here..

Example 3: Solve x² – 4x + 4 = 0 Easy to understand, harder to ignore..

1. Calculate the discriminant:
   b² – 4ac = (–4)² – 4(1)(4) = 16 – 16 = 0
2. Apply the quadratic formula:
   x = [–b ± √(0)]/(2a) = [4 ± 0]/2 = 2

Here, x = 2 is a repeated root. The graph of this equation is a parabola that touches the x-axis at one point (a tangent) Less friction, more output..

Example 4: Solve 3x² + 6x + 3 = 0.

1. Simplify by dividing all terms by 3:
   x² + 2x + 1 = 0
2. Recognize this as a perfect square:
   (x + 1)² = 0
3. Solve for x:
   x + 1 = 0 → x = –1

This quadratic equation has one solution because its discriminant is zero, resulting in a single intersection with the x-axis.

Systems of Equations: One Point of Intersection

Systems of equations involve two or more equations with the same variables. A system has exactly one solution when the equations intersect at a single point. This occurs when the lines are neither parallel nor coinciding Simple, but easy to overlook. Turns out it matters..

Example 5: Solve the system:

1. y = 2x + 1
2. y = –x + 4

Step 1: Substitute the expression for y from the first equation into the second:
2x + 1 = –x + 4

Step 2: Solve for x:
2x + x = 4 – 1
3x = 3
x = 1

Step 3: Substitute x = 1 into either equation to find y:
y = 2(1) + 1 = 3

The solution is (1, 3). Graphically, the lines intersect at this single point Not complicated — just consistent..

Example 6: Solve the system:

1. 2x + y = 5
2. x – y = 1

Step 1: Add the equations to eliminate y:
(2x + y) + (x – y) = 5 + 1
3x = 6
x = 2

Step 2: Substitute x = 2 into the second equation:
2 – y = 1 → y = 1

The solution is **(