Solve systems of equations by graphing

A system of two linear equations represents two lines whose common point, if one exists, gives the ordered pair satisfying both equations; graphing makes this shared relationship visible and distinguishes one solution, no solution for distinct parallel lines, and infinitely many solutions for coincident lines. The learner interprets intersection coordinates, including reasonable approximations when graphs do not show exact values, rather than treating each line’s intercept or slope as the system’s solution. The scope is limited to two-variable linear systems, not nonlinear or higher-dimensional systems.

Detailed Explanation: Solve systems of equations by graphing

A system of equations is a pair of equations that must both be true. When you graph the two equations, the solution is the point where the two lines intersect.

Consider the system

\begin{cases} y=2x+1\\ y=-x+7 \end{cases}

1. Graph the first equation

For $y=2x+1$ :

The $y$ -intercept is $1$ , so plot $(0,1)$ .
The slope is $2=\frac{2}{1}$ , so move up $2$ units and right $1$ unit.

This gives points such as

(0,1),\ (1,3),\ (2,5).

Draw a straight line through these points.

2. Graph the second equation

For $y=-x+7$ :

The $y$ -intercept is $7$ , so plot $(0,7)$ .
The slope is $-1=\frac{-1}{1}$ , so move down $1$ unit and right $1$ unit.

This gives points such as

(0,7),\ (1,6),\ (2,5).

Draw a straight line through these points.

3. Find the intersection

The two lines meet at

(2,5).

Therefore, the solution to the system is

\boxed{(2,5)}.

This means $x=2$ and $y=5$ make both equations true. Check:

5=2(2)+1

and

5=-2+7.

Both statements are true, so $(2,5)$ is the solution.

When graphing a system:

Lines that cross once have one solution.
Distinct parallel lines have no solution.
Lines that lie exactly on top of each other have infinitely many solutions.
The solution is always the shared point or points, not a line’s individual intercept or slope.

Learn by doing: Solve systems of equations by graphing

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Linear Equations - Find Intersection (Decimal) - Two Linear Equations

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