Section 5.1: Solving Systems of Linear Equations by Graphing

Property

A pair of linear equations with the same variables is called a system of linear equations. When we consider two equations together, we often use the same variables for both equations, like this:
$y = 6x + 1000$
$y = 2x + 1200$

Examples

• A gym offers two plans. Plan A: $y = 20x + 50$. Plan B: $y = 30x + 20$. This is a system where $y$ is the total cost for $x$ months.
• Two competing internet providers have different pricing. Company 1: $C = 10t + 100$. Company 2: $C = 15t + 50$. This system compares the cost $C$ over $t$ months.
• A student is saving money. Account 1: $A = 5w + 40$. Account 2: $A = 10w + 10$. This system models the amount $A$ in each account after $w$ weeks.

Explanation

Think of a system as two related math stories (equations) that use the same characters (variables). We analyze them together to find a shared outcome or a point where both are true simultaneously.

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

The solution of a system satisfies both equations in the system, so the point that represents the solution must lie on both graphs. It is the intersection point of the two lines described by the system. Thus, we can solve a system of equations by graphing the equations and looking for the point where the graphs intersect.

Examples

• To solve the system $y = x + 2$ and $y = -x + 4$, graph both lines. They intersect at the point $(1, 3)$. Therefore, the solution is $(1, 3)$.
• Consider the system $y = 2x$ and $y = 3x - 1$. Graphing these lines reveals they cross at $(1, 2)$. This ordered pair is the solution to the system.
• For the system $y = -2x + 5$ and $y = x - 1$, plotting both lines shows an intersection at $(2, 1)$. This means $x=2$ and $y=1$ is the solution.

Explanation

Imagine each equation is a path. The solution to the system is the treasure buried at the exact spot where the two paths cross. The coordinates of this intersection point will solve both equations.