Solving Systems of Equations by Graphing

Estimate the solution of a system of equations by graphing the equations on the same coordinate plane.
The ordered pair for the point of intersection of the graphs is the solution to the system.
To check the ordered pair, replace $x$ and $y$ with the values from ordered pair in each equation. If both sentences are true, the ordered pair is the solution to the system of equations.
Example: Solve the system of equations by graphing: $y = -2x - 3$ and $y = 2x + 5$ . The solution is $(-2, 1)$ .

If a linear equation is not in slope-intercept form, use properties of equality to rewrite the equation.
Example: Rewrite $4x + 2y = 16$ in slope-intercept form.
- $4x + 2y = 16$
- $2y = 16 - 4x$ (Subtraction Property of Equality)
- $y = 8 - 2x$ (Division Property of Equality)
- $y = -2x + 8$ (Commutative Property)

Some systems of equations have no solution.
If the graphs of the lines are parallel, they never intersect, and there is no solution.
Example: The system $y = 2x + 15$ and $y = 2x$ has no solution because the lines are parallel.

Some systems of equations have infinitely many solutions.
If the graphs of the lines are the same, they intersect at every point, and there is an infinite number of solutions.
Example: The system $-3x + y = 5$ and $y = 3x + 5$ has an infinite number of solutions because the lines are the same.