Junior High 2

Linear Functions

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Concept Explanation

A linear function describes a **constant rate of change**.

If x increases by 1, y always changes by the same amount — that amount is the **slope (a)**.

Where the line crosses the y-axis is the **y-intercept (b)**.

Think of it like a taxi meter: - The base fare is **b** (the y-intercept) - Every kilometer costs **a** yen more (the slope) - Total fare: y = ax + b

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

Solution

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.

Solution

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.

Solution

y = -x + 5

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

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