Master Linear Functions in 7 Days

A taxi charges a base fare of 500 yen, plus 100 yen per kilometer. Express the total fare y for x kilometers.

y = 100x + 500

The rate of change is 100 yen/km (slope a = 100). The starting cost is 500 yen (intercept b = 500). So the equation is y = 100x + 500.

For y = 2x + 3, find the value of y when x = 4.

Substitute x = 4 into y = 2x + 3:
y = 2(4) + 3
y = 8 + 3
y = 11

We replace x with 4 and compute step by step. Each step should show the calculation clearly.

Write the equation of a linear function with slope -1 and y-intercept 5.

y = -x + 5

Slope a = -1, intercept b = 5. Plug directly into y = ax + b.

Draw the graph of y = -2x + 4. Identify the y-intercept and one other point.

y-intercept: (0, 4)
When x = 1: y = -2(1) + 4 = 2 → point (1, 2)
When x = 2: y = -2(2) + 4 = 0 → point (2, 0)
Plot these points and draw a straight line through them.

Start at b = 4 on the y-axis. The slope is -2, so for every 1 step right, go 2 steps down.

Find the equation of the line passing through (0, 3) and (2, 7).

Let the equation be y = ax + b.

Find the slope:
a = (7 - 3) / (2 - 0) = 4 / 2 = 2

Since the line passes through (0, 3), b = 3.

Therefore y = 2x + 3

When one point is on the y-axis (x = 0), b is simply the y-coordinate of that point.

Find the equation of the line passing through (-1, 0) and (0, 2).

Let the equation be y = ax + b.

Find the slope:
a = (2 - 0) / (0 - (-1)) = 2 / 1 = 2

From point (0, 2): b = 2

Therefore y = 2x + 2

The point (0, 2) tells us b = 2 directly. Then we just need the slope.