Linear functions & equations

Japanese school year: Junior high 2

What you learn

You will learn that two-variable linear equations represent straight lines, and the solution to a system corresponds to their intersection point. This topic bridges algebraic calculations with geometric visualizations to deepen mathematical reasoning. Prior mastery of graphing linear functions and solving simultaneous equations is recommended.

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Key points

Solving the linear equation $ax + by = c$ ( $b \neq 0$ ) for $y$ reveals the slope $-\frac{a}{b}$ and the $y$ -intercept $\frac{c}{b}$ , allowing you to plot it as a line.

ax + by = c \quad (b \neq 0) \iff y = -\frac{a}{b}x + \frac{c}{b}

This indicates that the solution $(x, y)$ to a system of linear equations corresponds exactly to the coordinates of the intersection point of the two lines.

\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}

These equations represent lines parallel to the coordinate axes. The equation $x = k$ is a vertical line parallel to the $y$ -axis, and $y = l$ is a horizontal line parallel to the $x$ -axis.

x = k, \quad y = l

Choose a set to practice.

Practice 1Coming soon