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

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 \(7 \neq 5\), 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.