What does "solution" mean?

Question

What does it mean for a point to be a solution to a linear equation?

For example, if I say, "(2,5) is a solution to the equation y = 2x + 3." how could you check my claim?

Answer 1

The term "solution" generally refers to an answer to a problem or a means of resolving an equation. In the context of a mathematical equation, a solution is a set of values that satisfies the equation.

When we say that a point is a solution to a linear equation, we mean that the coordinates of that point, when substituted into the equation, make the equation true. This indicates that the point lies on the line defined by the equation.

For example, the equation $y=2x+3$ represents a line in the Cartesian coordinate system. A point $(x,y)$ is a solution to this equation if, when we substitute the $x$ value into the equation, we obtain the corresponding $y$ value.

To check if the point $(2,5)$ is a solution to the equation $y=2x+3$:

1. Substitute $x=2$ into the equation: $$y=2(2)+3$$

2. Calculate the right side: $$y=4+3=7$$

3. The calculated $y$ value is 7, while the $y$ value from the point $(2,5)$ is 5.

Since $5\ne 7$, this means that the point $(2,5)$ is not a solution to the equation $y=2x+3$.