Definition of a Solution to a System
This Grade 6 algebra skill from Yoshiwara Elementary Algebra defines a solution to a system of linear equations. Students learn that a solution is an ordered pair (x, y) that satisfies both equations simultaneously, and corresponds to the intersection point of the two lines on a graph.
Key Concepts
Property A solution to a system of equations is an ordered pair $(x, y)$ that satisfies each equation in the system simultaneously.
To check whether an ordered pair is a solution, substitute the coordinates into each equation to verify that they result in true statements.
Examples Verifying a Solution: Is $(3, 7)$ a solution to the system $y = 2x + 1$ and $y = 4x 5$? Check equation 1: $7 = 2(3) + 1$ becomes 7 = 7 (True). Check equation 2: $7 = 4(3) 5$ becomes 7 = 7 (True). Yes, it is the solution. The "One Line" Trap: Given the system $y = 2x + 1$ and $y = x + 4$, the point $(2, 5)$ lies on the first line but not the second line because $5 \neq 2 + 4$. Therefore, it is NOT a solution to the system.
Common Questions
What is a solution to a system of equations?
A solution to a system of two equations is an ordered pair (x, y) that makes both equations true at the same time.
How do you verify a solution to a system of equations?
Substitute the x and y values into both equations. If both equations are true, the ordered pair is a solution.
What does the solution represent on a graph?
The solution is the point where the two lines intersect. If the lines are parallel (no intersection), the system has no solution.
Can a system of equations have more than one solution?
If two equations represent the same line, every point on the line is a solution—infinitely many solutions. Distinct intersecting lines have exactly one solution.
Where is the definition of a solution to a system taught?
Definition of a solution to a system is covered in the Yoshiwara Elementary Algebra textbook for Grade 6.