In the realm of algebra, an equation that possesses only one solution is typically classified as a linear equation. More specifically, this happens when the equation is of the form:
ax + b = c
where:
- a is a non-zero constant, which guides the slope of the line.
- b and c are constants that adjust the positioning of the line along the y-axis.
When this equation is graphed, it forms a straight line. The important point to note is that if the line intersects the x-axis at a point, that point represents the single unique solution to the equation. For example:
2x + 4 = 10
To find the solution, we can rearrange it:
2x = 10 - 4
x = 3
Thus, the solution here is unique – x equals 3.
Another scenario where an equation has only one solution is when it is a quadratic equation that touches the x-axis at exactly one point. This occurs when the discriminant of the equation (part of the quadratic formula used to find the roots) is equal to zero. For example:
x² - 4x + 4 = 0
By factoring, we have:
(x - 2)(x - 2) = 0
This implies that x = 2 is the single solution.
Ultimately, both linear equations (when they have a slope) and specific quadratic equations (when they exhibit a perfect square scenario) provide frameworks in which we can find transformations leading to unique solutions.