To solve a system of equations by graphing, the goal is to graph each equation on the same coordinate plane and find the point(s) where the graphs intersect. The coordinates of the intersection point(s) are the solution(s) to the system of equations.

Steps for Solving a System of Equations by Graphing:

1. Rewrite each equation in slope-intercept form ($y = mx + b$, where $m$ is the slope and $b$ is the y-intercept) if they are not already in that form.
2. Graph both equations on the same coordinate plane.
3. Identify the point of intersection. The point where the two lines meet is the solution to the system.
   • If the lines intersect at a single point, the system has one unique solution.
   • If the lines are parallel (no intersection), the system has no solution.
   • If the lines are the same (coincident), the system has infinitely many solutions.

Example:

Solve the following system of equations by graphing: $y = 2x + 1$ $y = -x + 4$

1. Graph the first equation: The slope is 2 and the y-intercept is 1. Plot the point $(0, 1)$, then use the slope (rise 2, run 1) to plot additional points.
2. Graph the second equation: The slope is -1 and the y-intercept is 4. Plot the point $(0, 4)$, then use the slope (fall 1, run 1) to plot additional points.
3. Find the intersection: The lines intersect at the point $(1, 3)$.

Thus, the solution to the system of equations is $x = 1$ and $y = 3$.

Would you like further details on graphing this, or perhaps a different example?

5 Related Questions:

1. What happens if the system has no solution or infinitely many solutions?
2. How can you determine the number of solutions just by looking at the equations?
3. How do you handle systems of inequalities by graphing?
4. What are alternative methods to solve systems of equations (besides graphing)?
5. How does the slope-intercept form help in graphing systems?

Tip: Always check your graph carefully for accuracy by verifying the slope and y-intercept values when plotting points.