Lesson 15: Solving Systems of Equations by Graphing

New Concept

A system of equations is a collection of two or more equations containing two or more of the same variables.

What’s next

Next, you’ll learn the most visual way to find a solution: graphing the equations to see where they intersect.

A system of equations is a collection of two or more equations containing two or more of the same variables. A linear system contains two linear equations in two like unknowns. Solutions of systems are ordered pairs, representing the point of intersection.

The system $y = 3x - 2$ and $y = -x + 6$ has two equations with variables $x$ and $y$. The solution is where they meet.
For variables $a$ and $b$, a system could be $2a + b = 7$ and $a - b = 2$.
A cost system might be $C = 25 + 5d$ and $C = 10d$, where $d$ is days and $C$ is cost.

Think of it like a treasure hunt with two different maps! Each equation is a map drawing a line. The treasure, or the solution, is the one spot where the lines cross. You're looking for the single coordinate pair $(x, y)$ that makes both equations true. It's the ultimate point of agreement between two mathematical stories.

Both of the equations are graphed on the same coordinate grid. The coordinates of the point where the lines intersect, or cross, give the solution.

To solve $y=x+3$ and $y=-x+1$, graph both lines to see they intersect at the solution point $(-1, 2)$.
For the system $y = 2x - 2$ and $y = \frac{1}{2}x + 1$, graphing reveals the solution is the intersection point $(2, 2)$.
Graphing $y=4x$ and $y=x+6$ shows the lines cross at $(2, 8)$, which is the system's only solution.

Imagine you and a friend are walking separate straight paths on a giant city grid. The solution to the system is simply the corner where your paths cross! By drawing both lines on the same graph, you can visually pinpoint the exact coordinates $(x, y)$ of your meeting point. That single point is the only spot on both of your paths.

An inconsistent system is one that has no solution. When you graph the linear equations from an inconsistent system, you will see two parallel lines. Since parallel lines have the same slope but different intercepts, they run alongside each other forever and never cross. Therefore, no ordered pair $(x, y)$ can satisfy both equations at the same time.

The system $y = 2x + 3$ and $y = 2x - 1$ is inconsistent. The lines have the same slope ($2$) but different y-intercepts, so they are parallel.
For the system $y - 4x = 6$ and $2y = 8x - 10$, simplifying gives $y=4x+6$ and $y=4x-5$. These parallel lines will never intersect.

Imagine an inconsistent system as a pair of parallel train tracks. They run in the same direction forever but will never, ever meet. Because there's no intersection point, there's no solution that can make both equations happy. They are mathematically destined to never agree!

A dependent system occurs when every solution to one equation is also a solution to the other, resulting in an infinite number of solutions. When you graph the equations of a dependent system, you will find they are the exact same line. This is a special type of consistent system where the equations are just different forms of each other.

The system $y = 3x + 2$ and $3y = 9x + 6$ is dependent. The second equation is just the first one multiplied by three, representing the same line.
For $4x + 2y = 8$ and $y = -2x + 4$, simplifying the first equation also gives $y = -2x + 4$. The system is dependent with infinite solutions.

Think of a dependent system as a secret identity! Bruce Wayne and Batman are the same person, just in different outfits. The equations look different, but when you graph them, they trace the exact same line. Every point on that line is a solution!