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

Question

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

Answer 1

A point $(x,y)$ is said to be a solution to a linear equation if, when the coordinates of the point are substituted into the equation, the equation holds true. This means that the left-hand side of the equation should equal the right-hand side when you plug in the values of $x$ and $y$.

In your example, the point $(2,5)$ is claimed to be a solution to the equation $y=2x+3$. To check this claim, follow these steps:

1. Identify the values of $x$ and $y$ from the point. Here, $x=2$ and $y=5$.

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

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

4. Compare the calculated value of $y$ with the $y$ from the point $(2,5)$. We found $y=7$ from the equation, but the point gives $y=5$.

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

In summary, to verify if a point is a solution to a linear equation, substitute the $x$ value into the equation and check if the resulting $y$ value matches the $y$ coordinate of the point. If they match, the point is a solution; if they do not, it is not a solution.