Solving systems by graphing
Solve systems of equations by graphing in Grade 9 algebra. Plot both lines on a coordinate plane, find the intersection point, and verify the solution satisfies both original equations.
Key Concepts
Property If a system of linear equations has one solution, the solution is the common point or the point of intersection of their graphs. Explanation Solving a system by graphing is like a treasure hunt on a map! Each equation is a line showing a path. The solution is the treasure, and it's buried at the exact spot where the two lines cross. All you have to do is graph both lines on the same coordinate plane and find their point of intersection, $(x, y)$. Examples To solve the system $y = x + 1$ and $y = x + 5$, graph both lines. You'll see they intersect at the point $(2, 3)$, which is the solution. Solve the system $y = 2x$ and $x + y = 6$. Rewrite the second equation as $y = x + 6$. Graphing both lines reveals they intersect at the solution, $(2, 4)$. Check the solution $(2, 4)$ for $y = 2x$ and $y = x + 6$. Substitute into both: $4 = 2(2) \rightarrow 4=4 \text{ (True)}$ and $4 = 2 + 6 \rightarrow 4=4 \text{ (True)}$. The intersection point is correct!
Common Questions
How do you solve a system of equations by graphing?
Graph both equations on the same coordinate plane. The solution is the intersection point. Read the (x, y) coordinates and verify by substituting into both original equations.
What do parallel lines mean for a system of equations?
Parallel lines have no intersection, so the system has no solution. The equations are inconsistent — no x and y satisfies both simultaneously.
What does it mean when both equations graph as the same line?
The same line means the equations are dependent — every point on the line is a solution, giving infinitely many solutions.