Summary MPM2D (Grade 10 Math) – Linear Systems Notes & Worked Examples (Ontario Curriculum)

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​