Lesson 3: Systems of Linear Equations

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. Convert equations to slope-intercept form ($y = mx + b$) by isolating $y$.
2. Graph the first equation by plotting the y-intercept at $(0, b)$ and using the slope ($m$) to find the next point.
3. Graph the second equation on the same coordinate system.
4. Identify the point of intersection, which is the solution, and check it in both original equations.

Examples

• Converting First: Convert $3x + 2y = 8$ to slope-intercept form. Isolate $y$: $2y = -3x + 8$, so $y = -\frac{3}{2}x + 4$.
• Graphing to Solve: To solve the system $y = x + 1$ and $y = -x + 3$, we graph both lines. They intersect at the point $(1, 2)$. Checking this point in both equations confirms it is the solution.
• Special Lines: Remember that a horizontal line ($y = c$) has a slope of 0, and a vertical line ($x = c$) has an undefined slope. They are always perpendicular to each other.

Explanation

Many linear equations are not initially written in slope-intercept form, making it difficult to identify the slope and y-intercept directly. Once you use algebra to isolate $y$, graphing is like a simple treasure hunt. The solution to a system is the exact point where the two treasure paths cross.