1.10.3 Writing and Graphing Equations in Standard Form

Extending Your Thinking

Standard Form of Linear Equation

A linear equation is in standard form when it is written $Ax+By=C$.

Most people prefer to have $A$, $B$, and $C$ be integers and $A\ge 0$ when writing a linear equation in standard form, although it is not strictly necessary.

• Linear equations have infinitely many solutions.
• For every number that is substituted for $x$ there is a corresponding $y$ value.
• This pair of values is a solution to the linear equation and is represented by the ordered pair $(x,y)$.
• When we substitute these values of $x$ and $y$ into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.

Solution of a Linear Equation in Two Variables

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.

Linear equations have infinitely many solutions. We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.

A graph is a visual representation of all the solutions of the equation. It is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to that equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation. Points not on the line are not solutions!

When a linear equation in standard form is solved for $y$, it is then easy to pick out the slope of the line and where the graph of the line crosses the $y$–axis.