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Line through two points

Proportionality & linear functions

Japanese school year: Junior high 2

What you learn

Learn how to calculate the slope from two points with different coordinates to find the linear function whose graph is the line passing through them. This technique is useful for establishing linear models from observational data. Familiarity with slopes, intercepts, and systems of equations ensures smooth progress.

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Key points

This formula determines the equation of a line when its slope aa and the coordinates of one point (x1,y1)(x_1, y_1) on the line are known.

y−y1=a(x−x1)y - y_1 = a(x - x_1)

This formula directly finds the equation of a line passing through two distinct points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), with the fractional part calculating the slope between them.

y−y1=y2−y1x2−x1(x−x1)(x1≠x2)y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \quad (x_1 \neq x_2)

This method finds the slope aa and yy-intercept bb by substituting two given points into y=ax+by = ax + b and solving the resulting system of linear equations.

{ax1+b=y1ax2+b=y2\begin{cases} ax_1 + b = y_1 \\ ax_2 + b = y_2 \end{cases}

Choose a set to practice.