Lesson 4: Linear Equations

Property

An equation of the form $Ax + By = C$, where $A$ and $B$ are not both zero, is called a linear equation in two variables.
A linear equation is in standard form when it is written $Ax + By = C$. Linear equations have infinitely many solutions.
An ordered pair $(x, y)$ is a solution of the linear equation $Ax + By = C$, if the equation is a true statement when the $x$- and $y$-values of the ordered pair are substituted into the equation.

Examples

• The equation $5x + 2y = 10$ is a linear equation in two variables, with $A=5$, $B=2$, and $C=10$.

• The equation $y = 4x - 3$ is a linear equation. It can be rewritten in standard form as $4x - y = 3$.

Property

To find a solution to a linear equation with two variables, follow these steps:

1. Choose any value for one variable (e.g., $x$).
2. Substitute this value into the equation.
3. Solve the equation for the other variable (e.g., $y$).

The resulting pair of values $(x, y)$ is a solution to the equation.

Property

The graph of a linear equation $Ax + By = C$ is a straight line. Every point on the line is a solution of the equation. Every solution of this equation is a point on this line.

Examples

• Is the point $(2, 7)$ a solution to the equation $y = 3x + 1$? Yes, because substituting the values gives $7 = 3(2) + 1$, which simplifies to $7 = 7$. The point is on the line.

• Is the point $(1, 3)$ a solution to the equation $y = 3x + 1$? No, because substituting the values gives $3 = 3(1) + 1$, which simplifies to $3 = 4$. This is false, so the point is not on the line.