Middle school maths, year 2 · unit 6
Linear Functions and Equations
Unit 12 solved simultaneous equations by calculation, and unit 13 drew expressions as straight lines. They are the same story. Plot every solution of an equation and you get a line, and where two lines cross is the solution of the pair. We start by plotting points.
Plot every solution and you get a line
A board to adjustThe x and y that make x + y = 4 true go on for ever. Drag the gold handle on the x-axis to plot the solutions one by one.
Lines parallel to the axes
A board to adjustWhat does the graph of x = 3 look like? Drag the handles to move the vertical and horizontal lines.
Where they meet is the solution
A board to arrangeThe two equations of a pair are drawn as two lines. Drag the pin onto the point where the lines cross.
Meet, never meet, or coincide
A board to adjustIf the crossing point is the solution, what about two lines that never cross? Drag the two handles on the gold line.
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- Plot every solution of a linear equation in two unknowns, ax + by + c = 0, and you get a straight line. Solved for y, it is the same graph as a linear function y = ax + b.
- The graph of x = p is a vertical line parallel to the y-axis; the graph of y = q is a horizontal line parallel to the x-axis.
- The solution of a pair of simultaneous equations is the coordinates of the point where the two lines cross.
- Different slopes: they meet at one point (one solution). Same slope, different y-intercepts: parallel (no solution). Same slope and same y-intercept: they coincide (endlessly many solutions).