Lesson 1: Introduction to Systems and Solving by Graphing

Property

When two or more linear equations are grouped together, they form a system of linear equations. Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair $(x, y)$. To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

Examples

• Is $(2, 3)$ a solution to the system $\begin{cases} 3x - y = 3 \\ x + 2y = 8 \end{cases}$? For the first equation, $3(2) - 3 = 3$ is true. For the second, $2 + 2(3) = 8$ is true. Since it makes both true, $(2, 3)$ is a solution.
• Is $(-1, 5)$ a solution to the system $\begin{cases} 5x + y = 0 \\ 2x + y = 4 \end{cases}$? For the first equation, $5(-1) + 5 = 0$ is true. For the second, $2(-1) + 5 = 3 \neq 4$ is false. Therefore, $(-1, 5)$ is not a solution.
• Is $(4, -2)$ a solution to the system $\begin{cases} x + 3y = -2 \\ -2x - 5y = -2 \end{cases}$? For the first equation, $4 + 3(-2) = -2$ is true. For the second, $-2(4) - 5(-2) = -8 + 10 = 2 \neq -2$ is false. Therefore, $(4, -2)$ is not a solution.

Explanation

Think of a system's solution as a secret meeting point. It's the single ordered pair $(x, y)$ that exists on both lines at the same time. If a point only satisfies one equation, it hasn't arrived at the right spot.

Property

To solve a system of linear equations by graphing, follow these steps:

1. Graph the first equation.
2. Graph the second equation on the same rectangular coordinate system.
3. Determine whether the lines intersect, are parallel, or are the same line.
4. Identify the solution to the system. If the lines intersect, the point of intersection is the solution. Check the point in both equations to verify.

Examples

• Solve the system $\begin{cases} y = x + 2 \\ y = -x + 4 \end{cases}$ by graphing. The lines intersect at the point $(1, 3)$. Checking this point: $3=1+2$ (true) and $3=-1+4$ (true). The solution is $(1, 3)$.
• Solve the system $\begin{cases} x + y = 6 \\ 2x - y = 3 \end{cases}$ by graphing. The lines intersect at $(3, 3)$. Checking this point: $3+3=6$ (true) and $2(3)-3=3$ (true). The solution is $(3, 3)$.
• Solve the system $\begin{cases} y = 2 \\ x + y = 5 \end{cases}$ by graphing. The line $y=2$ is horizontal. The line $x+y=5$ intersects it at $(3, 2)$. The solution is $(3, 2)$.

Explanation

Graphing turns algebra into a visual treasure hunt. The two lines are paths, and the solution is the 'X' that marks the spot where they cross. This single point of intersection is the only ordered pair that satisfies both equations.