Linear Equations and Slope: Vocabulary Flashcards

General form of a linear equation

The general form involves two variables x and y, with fixed real constants A, B, C. The common representation is the linear equation in two variables:
Ax + By + C = 0.
This form describes a set of points (x, y) that satisfy the equation; geometrically, the graph is a straight line.
Key idea: with a linear equation, two points are enough to determine the line and its behavior.

The graph of a linear equation is a straight line.
To visualize the outputs (solutions), you can plot points that satisfy the equation:
- Pick an x value (the equation is defined for all real x), plug in to solve for y.
- Alternatively, set y to specific values to find corresponding x (leading to intercepts).
Intercepts (important test points):
- y-intercept: set x = 0, solve for y.
- x-intercept: set y = 0, solve for x.
Example equation: 4x - 3y = 12.
- With x = 0: -3y = 12 \,\Rightarrow\, y = -4, giving point (0,-4).
- With y = 0: 4x = 12 \,\Rightarrow\, x = 3, giving point (3,0).
A third point (test point) helps verify the line or understand slope more robustly:
- Choose x such that y is an integer; for example, x = 6 gives 4(6) - 3y = 12 \,\Rightarrow\, 24 - 3y = 12 \,\Rightarrow\, y = 4, giving point (6,4).
- This confirms consistency along the line: from $(0,-4)$ to $(3,0)$, and then to $(6,4)$, the slope remains the same.
Observing these points shows how the line behaves and confirms the equation.

Definition: slope m is the ratio of the change in y to the change in x:
m = rac{\Delta y}{\Delta x} = \frac{y2 - y1}{x2 - x1}.
For the points above, from $(0,-4)$ to $(3,0)$:
m = \frac{0 - (-4)}{3 - 0} = \frac{4}{3}.
Check with the next pair: from $(3,0)$ to $(6,4)$:
m = \frac{4 - 0}{6 - 3} = \frac{4}{3}.
A small slope corresponds to a shallow line (gentle tilt); a larger slope corresponds to a steeper line.
Interpretations of slope:
- Slope as rise over run: the vertical change relative to horizontal change.
- Slope as the angle with the x-axis: a smaller angle means a smaller slope; a larger angle means a larger slope.
Extreme cases:
- Horizontal line: slope m = 0 (no vertical change as x changes).
- Vertical line: slope is undefined (infinite); a vertical line has no finite slope.
- Therefore, slope is undefined for vertical lines.

Three basic forms mentioned:
- Standard form: Ax + By = C. (A, B, C are constants; rearranging gives slope-intercept form.)
- Slope-intercept form: y = mx + b. (m is the slope; b is the y-intercept.)
- A form that makes the function explicit in x, i.e., the slope-intercept form is often used because it directly shows the slope and y-intercept.
In the slope-intercept form, the coefficient of x is the slope: for y = mx + b, the slope is m. The constant b is the y-intercept, the point where the line crosses the y-axis.

Given equation:
0!x - 3y = 12. (This is the same as -3y = 12 with a hidden x-term of 0.)
Solve for y to obtain the slope-intercept form:
- Move the x-term to the other side (still 0 here):
  -3y = -0!\cdot x + 12.
- Divide by -3:
  y = \frac{4}{3}x - 4.
From this slope-intercept form:
- The slope is m = \frac{4}{3}.
- The y-intercept is b = -4, so the line crosses the y-axis at (0,-4).
Using slope-intercept form to graph:
- Start at the y-intercept $(0,-4)$.
- Use the rise/run corresponding to the slope: for every run of 3 units to the right, rise 4 units.
- Move from $(0,-4)$ to $(3,0)$ (rise 4, run 3).
- Move again to $(6,4)$, confirming alignment with the same slope.
Therefore, with two points e.g., $(0,-4)$ and $(3,0)$, you can draw the line, or you can use the slope-intercept form to plot directly.

Two points define a unique line; choose any two distinct points that satisfy the equation.
Common practice: use an intercept point and another point, then connect them.
As shown, using intercepts and a third test point helps verify the line and its slope.

Given a graph of a straight line, you can deduce an equation (to go from graph to equation).
Given an equation, you can graph it (to go from equation to graph).
The slope-intercept form makes it straightforward to read off the slope and y-intercept, which directly aids in graphing.
Intercepts provide quick, useful anchor points for plotting.
Remember the special cases: horizontal (slope 0) and vertical (undefined slope).