[Video Placeholder - Introduction to systems of equations]
Duration: 6:15
What are Systems of Equations?
A system of linear equations consists of two or more linear equations with the same variables. The solution is the point(s) where the equations intersect.
Methods for Solving Systems
1. Substitution Method
Solve one equation for one variable, then substitute into the other equation.
Solve the system:
y = 2x + 1
3x + y = 14
Step 1: Substitute y from first equation into second equation
3x + (2x + 1) = 14
Step 2: Combine like terms
5x + 1 = 14
Step 3: Solve for x
5x = 13 → x = 2.6
Step 4: Substitute back to find y
y = 2(2.6) + 1 = 6.2
Solution: (2.6, 6.2)
2. Elimination Method
Add or subtract equations to eliminate one variable.
Solve the system:
2x + 3y = 7
4x - y = 5
Step 1: Multiply second equation by 3 to match y coefficients
12x - 3y = 15
Step 2: Add to first equation
2x + 3y = 7
+ 12x - 3y = 15
14x = 22
Step 3: Solve for x
x = 22/14 ≈ 1.57
Step 4: Substitute to find y
4(1.57) - y = 5 → y ≈ 1.28
Solution: (1.57, 1.28)
3. Graphical Method
Graph both equations and find the intersection point.
Solve graphically:
y = -x + 4
y = 2x - 2
The lines intersect at (2, 2)
Solution: (2, 2)
Try it!
Solve the system:
3x + y = 10
2x - y = 5
Which method would you use?
x =
y =
Special Cases
No Solution (Parallel Lines)
y = 2x + 1
y = 2x - 3
These lines have the same slope (2) but different y-intercepts.