Graphs of Equations

While equations and functions are different, by drawing graphs of functions corresponding to equations, we can visually understand the characteristics and solutions of equations.

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Graphs of Linear Equations in Two Variables

Although equations and functions are different, if we can express an equation in the form $y=ax+b$ , we can draw it as a graph of a linear function. Let's say we have a linear equation in two variables:

$$2x + y = -1\quad-①$$

For this equation to be true, there are combinations of $x$ and $y$ such as:

$$(x, y)=(-2, 3),(-1, 1),(0, -1),(1, -3)$$

These solutions are combinations of $x$ and $y$ values, so we can represent them as coordinates on a graph. When we actually plot these points, they align on a straight line.

If we solve equation ① $2x+y=-1$ for $y$ , we get:

$$\begin{align*} 2x + y &= -1 \\ y &= -2x - 1 \quad-② \end{align*}$$

Equation ② is in the form $y=ax+b$ , so we can represent it as a graph of a linear function.

As we plot each solution, they align on a straight line, and the equation representing this line takes the form $y=ax+b$ . In other words, the straight line representing a linear equation in two variables represents the set of solutions to that equation.

Graphs of Equations in One Variable

Equations $x=k$ and $y=k$

What do the graphs of the following three equations look like?

① $y=2x+5$ has a slope of $2$ and a y-intercept of $5$ , so it looks like this. What about ② and ③?

$$\begin{align*} &y=2x+5\quad-①\\ &x=2\quad \quad \quad \,-②\\ &y=3\quad \quad \quad \,-③\\ \end{align*}$$

For ② $x=2$ , $x$ is always 2 regardless of $y$ , so when represented on a graph, it becomes a vertical line passing through the point $(2,0)$ and perpendicular to the x-axis. Similarly, for ③ $y=3$ , $y$ is always 3 regardless of $x$ , so it becomes a horizontal line passing through the point $(0,3)$ and parallel to the x-axis.

What if we treat the linear equation $ax+b=0$ as $y=ax+b$ ?

Next, let's consider the graph of the equation $2x=4$ . Let's transform this equation as follows:

$$\begin{align*} 2x &= 4 \\ 2x-4 &= 0\quad-③ \end{align*}$$

If we consider equation ③ as the case where $y=0$ for the linear function $y=2x-4$ , we can represent it as a graph of a linear function. Now, where does the solution $x=2$ of equation ③ appear on the graph?

The case where $y=0$ corresponds to the x-axis on the graph, so the intersection of the line $y=2x-4$ and the x-axis becomes the solution to equation ③.

Graphs of Systems of Equations

What the Intersection of Two Lines Means

We learned that when we represent a linear equation in two variables $ax+by=c$ on a graph, the line represents the set of solutions to the equation. So, what happens when we graph a system of two linear equations in two variables?

$$\begin{cases} \hspace{0.5em} x+y=7 \hspace{2em} - ①\\ \hspace{0.5em} 2x+y=10 \hspace{1em} - ② \end{cases}$$

Since there are two equations, two lines will appear, and each line represents the solutions to its respective equation. So...

The point of intersection of the two lines satisfies both equations ① and ②, which means the intersection point represents the solution to the system of equations.

There Are 3 Patterns in the Relationship Between Two Lines

When we draw a system of equations on a plane, two lines appear. These lines can intersect, be parallel, or overlap, so there are three patterns in total for the relationship between two lines.

We learned that the line represented by the graph of an equation is a collection of solutions to that equation. The intersection of two lines was the solution to the system of equations that simultaneously satisfies both equations. So, what can we say when two lines are parallel and don't intersect? Or what about when two lines completely overlap?

For the two lines representing sets of solutions:

• When they intersect, the intersection point satisfies both equations simultaneously, so the coordinates of the intersection point become the solution to the system of equations.
• When they are parallel, there are no coordinates that satisfy both equations simultaneously, so the system of equations has no solution.
• When they overlap, any point on the line satisfies both equations, so the system of equations has infinitely many solutions.