MPM2D (Grade 10 Math) - Linear Systems
Solving Linear Systems (Graphing)
- Linear system - two or more linear equations considered at the same time
- Point of intersection (POI) - point where two or more lines cross
- Solving a linear system - find values that satisfies all equations in the system
- Can have 1, 0, or infinite solutions
- Ordered pair (x, y) where lines intersect/touch (i.e. POI)
- Three ways to solve
1. Graphing
2. Substitution
3. Elimination
- Graphing (three ways)
1. Use slope and y-intercept → y = mx + b (linear equation form)
- Relationship between x (independent variable) and y (dependent variables)
- m = slope b = y-intercept
∆𝑦
→𝑚 = ∆𝑥
2. Use x- and y-intercepts of each line
3. Create a table of values for each equation
Example
Graph Slopes of Lines x- and y-intercepts Number of Solutions
Intersecting Different slopes Different unless POI 1 solution
y1 = 2x1 + 5 is on an axis (x or y) the lines intersect at
y2 = 4x2 exactly one point
Parallel and Distinct Same slope Different intercepts no solutions
the lines never touch
y1 = 3x1
y2 = 3x2 + 19
Parallel and Coincident Same slope Same intercepts infinite (∞) solutions
y1 = 5x1 the lines
y2 = 5x2 continuously touch
- Equations may seem different but
- Parallel and distinct - same slope, different y-intercepts → no solutions
- Parallel and coincident - same slope, same y-intercept → infinite (∞) solutions
Solving Linear Systems (Graphing)
- Linear system - two or more linear equations considered at the same time
- Point of intersection (POI) - point where two or more lines cross
- Solving a linear system - find values that satisfies all equations in the system
- Can have 1, 0, or infinite solutions
- Ordered pair (x, y) where lines intersect/touch (i.e. POI)
- Three ways to solve
1. Graphing
2. Substitution
3. Elimination
- Graphing (three ways)
1. Use slope and y-intercept → y = mx + b (linear equation form)
- Relationship between x (independent variable) and y (dependent variables)
- m = slope b = y-intercept
∆𝑦
→𝑚 = ∆𝑥
2. Use x- and y-intercepts of each line
3. Create a table of values for each equation
Example
Graph Slopes of Lines x- and y-intercepts Number of Solutions
Intersecting Different slopes Different unless POI 1 solution
y1 = 2x1 + 5 is on an axis (x or y) the lines intersect at
y2 = 4x2 exactly one point
Parallel and Distinct Same slope Different intercepts no solutions
the lines never touch
y1 = 3x1
y2 = 3x2 + 19
Parallel and Coincident Same slope Same intercepts infinite (∞) solutions
y1 = 5x1 the lines
y2 = 5x2 continuously touch
- Equations may seem different but
- Parallel and distinct - same slope, different y-intercepts → no solutions
- Parallel and coincident - same slope, same y-intercept → infinite (∞) solutions
