Grade 9 General Mathematics: Linear Functions and Equations Review

Flashcards covering key vocabulary, linear relationships, rate of change, slope calculations, slope-intercept conversions, and graph representations from Grade 9 General CA3 Term 1 review notes.

Rate of Change

The ratio that describes how much one quantity changes in relation to another quantity, calculated as $\frac{\Delta y}{\Delta x}$.

Linear Function Rate of Change (Example 3)

A constant rate of change of $-\frac{2}{3}$, confirming that the relationship shown in the table with coordinates $(11, -5)$, $(8, -3)$, $(5, -1)$, $(2, 1)$, and $(-1, 3)$ is linear.

Nonlinear Relationship (Example 4)

A relationship where the rate of change is not constant across data pairs (e.g., changing from $\frac{3}{7}$ to $\frac{2}{7}$), showing that the function is not linear.

Slope-Intercept Form

A linear equation written in the form $y = mx + b$, where $m$ is the slope of the line and $b$ is the $y$-intercept.

Slope-Intercept Form of $-10x + 2y = 12$

$y = 5x + 6$, obtained by adding $10x$ to both sides and dividing every term by $2$.

Slope-Intercept Form of $4y + 12x = 16$

$y = -3x + 4$, obtained by subtracting $12x$ from both sides and dividing every term by $4$.

Slope-Intercept Form of $-5x + 15y = -30$

$y = \frac{1}{3}x - 2$, obtained by adding $5x$ to both sides and dividing every term by $15$.

Slope-Intercept Form of $6x - 3y = -18$

$y = 2x + 6$, obtained by subtracting $6x$ from both sides and dividing every term by $-3$.

Slope-Intercept Form of $-2x - 8y = 24$

$y = -\frac{1}{4}x - 3$, obtained by adding $2x$ to both sides and dividing every term by $-8$.

Slope-Intercept Form of $-4x - 10y = -7$

$y = -\frac{2}{5}x + \frac{7}{10}$, obtained by adding $4x$ to both sides and dividing every term by $-10$.

Equation of the Line through $(0, -3)$ and $(2, 1)$

$y = 2x - 3$, which has a $y$-intercept of $-3$ and a slope of $m = \frac{1 - (-3)}{2 - 0} = 2$.

Equation of the Line through $(0, 2)$ and $(2, -4)$

$y = -3x + 2$, which has a $y$-intercept of $2$ and a slope of $m = \frac{-4 - 2}{2 - 0} = -3$.

Equation of the Line through $(0, -1)$ and $(2, -3)$

$y = -x - 1$, which has a $y$-intercept of $-1$ and a slope of $m = \frac{-3 - (-1)}{2 - 0} = -1$.

Slope Formula

The formula used to find the slope $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Slope of the Line through $(-1, 3)$ and $(4, 1)$

$-\frac{2}{5}$, calculated using $\frac{1 - 3}{4 - (-1)} = \frac{-2}{5}$.

Slope of the Line through $(-2, -5)$ and $(4, -5)$

$0$, indicating a horizontal line where there is zero vertical change ($\Delta y = 0$).

Slope of the Line through $(-3, 4)$ and $(-3, -2)$

Undefined, indicating a vertical line where there is zero horizontal change ($\Delta x = 0$), resulting in division by zero.

Graph of $5x + 2y = 8$

A line with a negative slope of $-\frac{5}{2}$, a $y$-intercept at $(0, 4)$, and an $x$-intercept at $(1.6, 0)$.

Graph of $4x + 9y = 27$

A line with a negative slope of $-\frac{4}{9}$, a $y$-intercept at $(0, 3)$, and an $x$-intercept at $(6.75, 0)$.

Graph of $y = 7$

A horizontal line with a slope of $0$ that crosses the $y$-axis at $(0, 7)$.

Graph of $y = -\frac{2}{3}$

A horizontal line with a slope of $0$ located below the $x$-axis, crossing the $y$-axis at $\left(0, -\frac{2}{3}\right)$.

Graph of $21 = 7y$

A horizontal line equivalent to $y = 3$, having a slope of $0$ and crossing the $y$-axis at $(0, 3)$.

Graph of $3y - 6 = 2x$

A line rewritten as $y = \frac{2}{3}x + 2$, with a positive slope of $\frac{2}{3}$, a $y$-intercept at $(0, 2)$, and an $x$-intercept at $(-3, 0)$.