Solve systems of linear equations graphically

A system of two linear equations in two variables is understood as a pair of lines whose common point represents an ordered pair satisfying both equations; the graphical solution is their intersection, interpreted exactly when possible and approximately when read from a graph. The reasoning distinguishes one solution (intersecting lines), no solution (distinct parallel lines), and infinitely many solutions (coincident lines), connecting graphical solutions to algebraic methods without extending to nonlinear or higher-dimensional systems.

Detailed Explanation: Solve systems of linear equations graphically

A system of two linear equations represents two lines on the same coordinate plane. The solution is the ordered pair where the lines intersect, because that point lies on both lines.

Example

Solve the system graphically:

\begin{cases} 2x+y=5\\ x-y=1 \end{cases}

1. Rewrite each equation in slope-intercept form, $y=mx+b$ .

For the first equation:

2x+y=5

Subtract $2x$ from both sides:

y=-2x+5

For the second equation:

x-y=1

Subtract $x$ from both sides:

-y=1-x

Multiply by $-1$ :

y=x-1

So graph these two lines:

y=-2x+5 \qquad\text{and}\qquad y=x-1

2. Plot points for each line.

For $y=-2x+5$ :

If $x=0$ , then $y=5$ , giving $(0,5)$ .
If $x=2$ , then $y=1$ , giving $(2,1)$ .

For $y=x-1$ :

If $x=0$ , then $y=-1$ , giving $(0,-1)$ .
If $x=2$ , then $y=1$ , giving $(2,1)$ .

Draw a straight line through each pair of points. The lines intersect at $(2,1)$ .

3. State the solution.

The graphical solution is

\boxed{(2,1)}

This means $x=2$ and $y=1$ satisfy both original equations:

2(2)+1=5

and

2-1=1.

When graphing a system, the possibilities are:

One solution: the lines intersect once.
No solution: the lines are distinct and parallel.
Infinitely many solutions: the two equations graph as the same line.

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Types of Solutions - Graph to Zero, One, or Infinite

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