Lesson 1: Estimate Solutions by Inspection

Property

A system of linear equations consists of two or more linear equations that share the same variables and are considered simultaneously.

The solution to a system of linear equations in two variables ($x$ and $y$) is any ordered pair $(x, y)$ that makes BOTH equations true at the same time. Geometrically, this solution is the exact point where the graphs of the two lines intersect.

Examples

• Example 1 (Verifying a Solution): Is the ordered pair $(2, 1)$ a solution to the system $x + y = 3$ and $2x - y = 3$?

Substitute $x=2$ and $y=1$ into both equations:
Equation 1: $2 + 1 = 3$ (True)
Equation 2: $2(2) - 1 = 3 \rightarrow 4 - 1 = 3$ (True)
Because both statements are true, $(2, 1)$ is the solution to the system.

• Example 2 (Not a Solution): Is $(-1, 3)$ a solution to the system $y - x = 4$ and $2x + y = 0$?

Equation 1: $3 - (-1) = 4 \rightarrow 4 = 4$ (True)
Equation 2: $2(-1) + 3 = 0 \rightarrow -2 + 3 = 0 \rightarrow 1 = 0$ (False)
Because it fails the second equation, $(-1, 3)$ is NOT a solution to the system.

Explanation

Think of a single linear equation as a road, and its solutions are all the locations on that road. A system of equations represents two roads on the same map. The "solution to the system" is the crossroads—the single intersection point $(x, y)$ where the two roads meet. This specific pair of numbers is the only location that exists on both roads at the exact same time.

Property

If two linear equations have the exact same slope (m), they will never intersect just once. You must check their y-intercepts (b) to determine the outcome:

• Different b (No Solution): The system has the same slope but different intercepts. They are parallel, so there is no solution.
• Same b (Infinite Solutions): The system has the same slope and the same y-intercept. They overlap everywhere, giving infinitely many solutions.

Examples

• No Solution: The system $y = 3x + 2$ and $y = 3x - 1$ has the same slope ($m=3$) but different $y$-intercepts. The lines are parallel, so there is no solution.
• Infinite Solutions: The system $x + y = 5$ and $2x + 2y = 10$ represents the same line, as the second equation is double the first. There are infinitely many solutions.

Explanation

When two lines have the same slope, they are traveling in the exact same direction at the exact same speed. If they start at different points on the y-axis, they will run parallel forever and never touch (zero solutions). But if they have the same slope AND start at the exact same y-intercept, they are actually a single line disguised as two different equations, meaning every point on the line is a shared solution (infinite solutions)!