Solution of a system of linear equations
Identify the solution of a system of linear equations as the ordered pair (x, y) where both equations are satisfied simultaneously, represented as a graph intersection in Grade 9 Algebra.
Key Concepts
Property A solution of a system of linear equations is any ordered pair that makes all the equations true. Explanation Think of a solution as a secret password! An ordered pair $(x, y)$ is the correct password only if it works for every single equation in the system. If it fails even one, the whole thing is wrong. To check, substitute the x and y values into each equation. If they all result in true statements, you've cracked the code! Examples Is $(2, 1)$ a solution for $2x + y = 5$ and $x y = 1$? Check: $2(2) + 1 = 5 $ ,$2 1=1$ Is $(4, 1)$ a solution for $x + y = 5$ and $x 2y = 3$? Check: $4 + 1 = 5 $ ,$4 2(1)=3$ Is $( 1, 3)$ a solution for $y = 2x + 1$ and $y = x + 4$? Check: $3 = 2( 1) + 1 $ ,$3= 1+4$.
Common Questions
What is the solution to a system of linear equations?
The solution to a system of linear equations is the ordered pair (x, y) that satisfies all equations in the system simultaneously. Graphically, it is the point where the lines intersect. A system can have one solution (lines intersect), no solution (parallel lines), or infinitely many solutions (same line).
How do you verify a solution to a system of equations?
Substitute the x and y values from your solution into each original equation. If both equations produce true statements when you plug in the values, then (x, y) is the correct solution. If either equation is false, recheck your work — the ordered pair must satisfy every equation.
What does it mean when a system has no solution?
A system has no solution when the equations represent parallel lines that never intersect. Parallel lines have the same slope but different y-intercepts. When solving algebraically, you will get a contradiction like 0 = 5, which has no solution, indicating the lines are parallel and the system is inconsistent.