Lesson 1: Solving Systems of Linear Equations by Graphing

Property

If a system of two linear equations has a unique solution, that solution is the ordered pair $(x, y)$ corresponding to the point of intersection of the graphs of the two equations.

Examples

Property

To solve a system of linear equations by graphing, follow these steps:

1. Graph the first equation.
2. Graph the second equation on the same rectangular coordinate system.
3. Determine whether the lines intersect, are parallel, or are the same line.
4. Identify the solution to the system. If the lines intersect, the point of intersection is the solution. Check the point in both equations to verify.

Examples

• Solve the system $\begin{cases} y = x + 2 \\ y = -x + 4 \end{cases}$ by graphing. The lines intersect at the point $(1, 3)$. Checking this point: $3=1+2$ (true) and $3=-1+4$ (true). The solution is $(1, 3)$.
• Solve the system $\begin{cases} x + y = 6 \\ 2x - y = 3 \end{cases}$ by graphing. The lines intersect at $(3, 3)$. Checking this point: $3+3=6$ (true) and $2(3)-3=3$ (true). The solution is $(3, 3)$.
• Solve the system $\begin{cases} y = 2 \\ x + y = 5 \end{cases}$ by graphing. The line $y=2$ is horizontal. The line $x+y=5$ intersects it at $(3, 2)$. The solution is $(3, 2)$.

Explanation

Graphing turns algebra into a visual treasure hunt. The two lines are paths, and the solution is the 'X' that marks the spot where they cross. This single point of intersection is the only ordered pair that satisfies both equations.

Property

To solve real-world problems using systems of equations by graphing:

1. Identify the two unknown quantities and assign variables.
2. Write two equations based on the given relationships.
3. Graph both equations on the same coordinate plane.
4. Find the intersection point and interpret the coordinates in context.

Examples

• Number Puzzle: Two numbers have a sum of 12 and a difference of 2. Graph the system $x + y = 12$ and $x - y = 2$. The intersection point $(7, 5)$ means the numbers are 7 and 5.
• Coffee Shop: Premium beans cost 12 per pound and regular beans cost 8 per pound. A customer buys 10 pounds total for 92. Graph the system $x + y = 10$ and $12x + 8y = 92$. The intersection point $(3, 7)$ means 3 pounds of premium and 7 pounds of regular beans were bought.
• Geometry: A rectangle has a perimeter of 20 inches, and the length is 4 inches more than the width. Graph the system $2L + 2W = 20$ and $L = W + 4$. The intersection point $(7, 3)$ means the length is 7 inches and the width is 3 inches.

Explanation

Real-world problems often involve two unknown quantities with two entirely different relationships between them (like counting total items vs. counting total money). Each relationship becomes its own equation. The intersection point of the graphed lines gives the unique values that satisfy both conditions simultaneously.